POPULAR LECTURES 
 
 ON 
 
 SCIENTIFIC SUBJECTS. 
 
 SECOND SERIES. 
 
New and Cheaper Edition, uniform. 
 
 HELMHOLTZ'S POPULAE LECTUEES on SCIENTIFIC 
 SUBJECTS, FIRST SERIES. Translated by E. ATKINSON, Ph.D. F.C.S. 
 with Introduction by Prof. TYNPALL. New and Cheaper Edition, with 
 51 Woodcuts. Crown 8vo. price 7s. 6d. 
 
 LIST of the LECTURES : 
 
 s- 
 S. 
 l- 
 
 1. On the Relation of Natural Science 
 
 to Science in General. Tran 
 lated byH. W. EVE, M.A. F.C 
 Head Master of University Co 
 lege School. 
 
 2. On Goethe's Scientific Researches. 
 
 Also translated by Mr. Eve. 
 
 3. On the Physiological Causes of 
 
 Harmony in Music. Translated 
 by A. J. ELLIS, M.A. F R.S. 
 
 4. Ice and Glaciers, Translated by 
 
 E. ATKINSON, Ph.D. F.C.S. Pro- 
 fessor of Experimental Science, 
 Staff College. 
 
 5. On the Interaction of the Natural 
 
 Forces. Translated by Professor 
 TYNDALL, LL.D. F.R.S. 
 
 6. The Recent Progress of the 
 
 Theory of Vision. Translated 
 by P. H. PYE-SMITH, B.A. M.D. 
 F.R.C.P. Guy's Hospital : 
 
 I. The Eye as an Optical 
 
 Instrument. 
 
 II. The Sensation of Sight. 
 III. The Perception of Sight. 
 
 7. On the Conservation of Force. 
 
 Translated by E. ATKINSON, 
 Ph.D. F.C.S. 
 
 8. On the Aim and P 
 
 sical Science. 
 WALTER FLIGHT, 
 Museum. 
 
 of Phy- 
 anslated by 
 ,Sc. British 
 
 HELMHOLTZ on the SENSATIONS of TONE as a Physio- 
 logical Basis for the Theory of Music. Translated, with the Aitthor's 
 sanction, from the Third German Edition, with Additional Notes and 
 an Additional Appendix, by ALEXANDER J. ELLIS, F.R.S. &c. 8vo. 
 price 365. 
 
 'It is hardly too much to say that this 
 volume far exceeds in value any and 
 every similar work.' ORCHESTRA. 
 
 ' The most important contribution 
 to the science of music which has at 
 any period been received from a single 
 source.' MUSICAL STANDARD. 
 
 ' The present book supersedes all 
 other treatises on the physics of 
 musical sound and the necessary 
 
 relations of this to systems of melody 
 and harmony.' 
 
 PALL MALL GAZETTE. 
 
 ' It is unnecessary for us to say 
 that this famous book will be wel- 
 comed alike by the 'physicist, the 
 acoustician, and the musician. It is 
 one of the most original works of the 
 second half of this century.' 
 
 QUARTERLY JOURNAL OF SCIENCE. 
 
 London, LONGMANS & CO. 
 
POPULAR LECTURES 
 
 ON 
 
 SCIENTIFIC SUBJECTS, 
 
 BY 
 
 H. HELMHOLTZ, 
 
 IT 
 
 PROFESSOR OF PHYSICS IX THE UNIVERSITY OF BERLIN 
 
 TRANSLATED BY 
 
 E. ATKINSON, PH.D. F.C.S. 
 
 PROFESSOR OF EXPERIMENTAL SCIENCE, STAFF COLLEGE. 
 
 B U A II Y 
 
 ^Y OF | 
 
 CAL1FOUN1A. 
 
 LONDON : 
 LONGMANS, GKEEN, AND CO. 
 
 1881. 
 
 All rights reserved. 
 

PBEFACE. 
 
 THE FAVOUR with which the first series of Professor 
 Helmholtz's Lectures has been received would justify, 
 if a justification were needed, the publication of the 
 present volume. 
 
 I have to express my acknowledgments to Pro- 
 fessor G. Croome Kobertson, 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 
 
VI PREFACE. 
 
 may be said to date from the year 1854, has brought 
 about so many alterations both in the form and in the 
 spirit of the regulations, that older members of the 
 University have been known to speak of the place 
 as so changed that they could scarcely recognise it. 
 Hence, in respect of this article, I have availed myself 
 of the liberty granted by Professor Helmholtz, and 
 have altogether omitted some passages, and have 
 slightly modified others, which would convey an erro- 
 neous impression of the present state of things. I 
 have also on these points consulted members of the 
 University on whose judgment I think I can rely. 
 
 In other articles, where the matter is of prime 
 importance, I have been anxious faithfully to repro- 
 duce the original ; nor have I in any such cases al- 
 lowed a regard for form to interfere with the plain 
 duty of exactly rendering the author's meaning. 
 
 E. ATKINSON. 
 
 PORTESBERY HlXL, CAMBERLEY : 
 
 Dec. 1880. 
 
CONTENTS. 
 
 rECTUBE 
 
 I. GUSTAV MAGNUS. IN MEMOEIAM .... 1 
 
 II. ON THE ORIGIN AND SIGNIFICANCE OF GEOMETRICAL 
 
 AXIOMS. ... . . . . . .27 
 
 in. ON THE RELATION OF OPTICS TO PAINTING . . 7$ 
 I. Form. . . . . . . j . .78 
 
 II. Shade . . - ".'.'-' 9* 
 
 in. Colour . . . HO 
 
 iv. Harmony of Colour . 124 
 
 IV. ON THE ORIGIN OF THE PLANETARY SYSTEM . . 139 
 
 Y. ON THOUGHT IN MEDICINE 199 
 
 VL ON ACADEMIC FREEDOM IN GERMAN UNIVERSITIES . 237 
 

 LIBRA H 
 
 UNIVKKSJTV OF 
 
 CALIFORNIA. 
 
 GUSTAV MAGNUS. 
 3n 
 
 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 Grustav 
 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 
 Grerman 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 I 
 
 B 
 
2 GUSTAV MAGNUS. 
 
 to pronounce a competent judgment on the principal 
 side of Magnus's activity, namely, the chemical. 
 
 Hence what remains for me to do is greatly re- 
 stricted. I shall scarcely venture to speak as the 
 biographer of Magnus, but only of what he was to us 
 and to science, to represent which is our allotted task. 
 
 His life was not indeed rich in external events 
 and changes ; it was the peaceful life of a man who, 
 free from the cares of outer circumstances, first as 
 member, then as leader of an esteemed, gifted, and 
 amiable family, sought and found abundant satisfaction 
 in scientific work, in the utilisation of scientific results 
 for the instruction and advantage of mankind. Hein- 
 rich Grustav Magnus was born in Berlin on May 2, 
 1802, the fourth of six brothers, who by their talents 
 have distinguished themselves in various directions. 
 The father, Johann Matthias, was chief of a wealthy 
 commercial house, whose first concern was to secure 
 to his children a free development of their individual 
 capacity and inclinations. Our departed friend showed 
 very early a greater inclination for the study of mathe- 
 matics and natural philosophy than for that of lan- 
 guages. His father arranged his instruction accor- 
 dingly, by removing him from the Werder Gymnasium 
 and sending him to the Cauer Private Institute, in 
 which more attention was paid to scientific subjects. 
 
 From 1822 to 1827 Magnus devoted himself en- 
 
GUSTAV MAGNUS. 3 
 
 tirely to the study of natural science at the University 
 of Berlin. Before carrying out his original intention 
 of qualifying as a professor of technology, he spent two 
 years with that object in travelling ; he remained with 
 Berzelius a long time in Stockholm, then with Du- 
 long, Thenard and Gay-Lussac in Paris. Unusually 
 well prepared by these means, he qualified in the 
 University of this place in technology, and afterwards 
 also in physics ; he was appointed extraordinary pro- 
 fessor in 1834, and ordinary professor in 1845, and so 
 distinguished himself by his scientific labours at this 
 time, that nine years after his habilitation, on January 
 27, 1840, he was elected a member of this Academy. 
 From 1832 until 1840 he taught physics in the 
 Artillery and Engineering School ; and from 1850 until 
 1856 chemical technology in the Gewerbe Institut. 
 For a long time he gave the lectures in his own house, 
 using his own instruments, which gradually developed 
 into the most splendid physical collection in existence 
 at that time, and which the State afterwards purchased 
 for the University. His lectures were afterwards given 
 in the University, and he only retained the laboratory 
 in his own house for his own private work and for that 
 of his pupils. 
 
 His life was passed thus in quiet but unremitting 
 activity; travels, sometimes for scientific or technical 
 studies, sometimes also in the service of the State, and 
 
 2 
 
4 G-USTAV MAGNUS. 
 
 occasionally for recreation, interrupted his work here 
 from time to time. His experience and knowledge of 
 business were often in demand by the State on various 
 commissions; among these may be especially men- 
 tioned the part he took in the chemical deliberations 
 of the Agricultural Board (Landes-Economie Colle- 
 gium), to which he devoted much of his time ; above 
 all to the great practical questions of agricultural 
 chemistry. 
 
 After sixty-seven years of almost undisturbed 
 health he was overtaken by a painful illness towards 
 the end of the year 1869. 1 He still continued his 
 lectures on physics until February 25, 1870, but dur- 
 ing March he was scarcely able to leave his bed, arfd he 
 died on April 4. 
 
 Magnus's was a richly endowed nature, which under 
 happy external circumstances could develop in its own 
 peculiar manner, and was free to choose its activity 
 according to its own mind. But this mind was so 
 governed by reason, and so filled, I might almost say, 
 with artistic harmony, which shunned the immoderate 
 and impure, that he knew how to choose the object of 
 his work wisely, and on this account almost always to 
 attain it. Thus the direction and manner of Magnus's 
 activity accorded so perfectly with his intellectual 
 nature as is the case only with the happy few among 
 1 Carcinoma recti. 
 
GUSTAV MAGNUS. 5 
 
 mortals. The harmonious tendency and cultivation of 
 his mind could be recognised in the natural grace of 
 his behaviour, in the cheerfulness and firmness of his 
 disposition, in the warm amiability of his intercourse 
 with others. There was in all this, much more than 
 the mere acquisition of the outer forms of politeness 
 can ever reach, where they are not illuminated by a 
 warm sympathy and by a fine feeling for the beautiful. 
 Accustomed from an early age to the regulated and 
 prudent activity of the commercial house in which he 
 grew up, he retained that skill in business which he 
 had so frequently to exercise in the administration of the 
 affairs of this Academy, of the philosophical faculty, and 
 of the various Government commissions. He retained 
 from thence the love of order, the tendency towards 
 the actual, and towards what is practically attainable, 
 even although the chief aim of his activity was an ideal 
 one. He understood that the pleasant enjoyment of 
 an existence free from care, ad 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 staking dis- 
 coveries, which might at once have made his name 
 celebrated. He was, on the contrary, a master of faith- 
 
6 GUSTAV MAGNUS. 
 
 ful, 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, Regnault 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. 
 
G-USTAV MAGNUS. 7 
 
 He invited me at that time (I myself would not 
 have ventured to propose it) to extend my experiments 
 on fermentation and putrefaction in new directions, 
 and to apply other methods, which required greater 
 means than a young army surgeon living on his pay 
 could provide himself with. At that time I worked 
 with him almost daily for about three months, and thus 
 gained a deep and lasting impression of his goodness, 
 his unselfishness, and his perfect freedom from scientific 
 jealousy. 
 
 By such a course he not only surrendered the ex- 
 ternal advantages which the possession of one of the 
 richest collections of instruments would have secured 
 an ambitious man against competitors, but he also bore 
 with perfect composure the little troubles and vexations 
 involved in the want of skill and the hastiness with 
 which young 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 
 the inner tendency of the man, which must not be 
 
8 GUSTAV MAGNUS. 
 
 neglected in estimating his value : he was not only an 
 investigator, he was also a teacher of science in the 
 highest and widest sense of the word. He did not wish 
 science to be confined to the study and lecture-room, 
 he desired that it should find its way into all conditions 
 of life. In his active interest for technology, in his 
 zealous participation in the work of the Agricultural 
 Board, this phase of his efforts was plainly reflected, as 
 well as in the great trouble he took in the preparation 
 of experiments, and in the ingenious contrivance of the 
 apparatus required for them* 
 
 His collection of instruments, which subsequently 
 passed into the possession of the University, and is 
 at present used by me as his successor, is the most 
 eloquent testimony of this. Everything is in the most 
 perfect order : if a silk-thread, a glass tube, or a cork, 
 are required for an experiment, one may safely depend 
 on finding them near the instrument. All the appa- 
 ratus which he contrived is made with the best means 
 at his disposal, without sparing either material, or the 
 labour of the workman, so as to ensure the success of 
 the experiment, and by making it on a sufficiently large 
 scale to render it visible as far off as possible. I recol- 
 lect very well with what wonder and admiration we 
 students sa<w him experiment, not merely because 
 all the experiments were successful and brilliant, but 
 because they, scarcely seemed to occupy or to disturb his 
 
GUSTAV MAGNUS. 9 
 
 thoughts. The easy and clear flow of his discourse 
 went on without interruption; each experiment came 
 in its right place, was performed quickly, without haste 
 or hesitation, and was then put aside. 
 
 I have already mentioned that the valuable collec- 
 tion of apparatus came into the possession of the 
 University during his lifetime. He specially wished 
 that what he had collected and constructed as appli- 
 ances in his scientific work should not be scattered and 
 estranged from the original purpose to which he had 
 devoted his life. With this feeling he bequeathed to 
 the University the rest of the apparatus of his labora- 
 tory, as well as his very rich and valuable library, and 
 he thus laid the foundation for the further development 
 of a Public Physical Institute. 
 
 It is sufficient in these few touches to have recalled 
 the mental individuality of our departed friend, so far 
 as the sources of the direction of his activity are to be 
 found. 
 
 Personal recollections will furnish a livelier image 
 to all those of you who have worked with him for the 
 last thirty years. 
 
 If we now proceed, to discuss the results of his 
 researches it will not be sufficient to read, through 
 and to estimate his academical writings. I have 
 already shown that a prominent part of his activity was 
 directed to his fellow-creatures. To this must be added, 
 
10 GUSTAV MAGNUS. 
 
 that he lived in an age when natural science passed 
 through a process of development, with a rapidity such 
 as never occurred before in the history of science. 
 But the men who belonged to such a time, and co- 
 operated in this development are apt to appear in 
 wrong perspective to their successors, since the best 
 part of their work seems to the latter self-evident, and 
 scarcely worthy of mention. 
 
 It is difficult for us to realise the condition of natural 
 science as it existed in Germany, at least in the first 
 twenty years of this century. Magnus was born in 
 1802; I myself nineteen years later; but when I go 
 back to my earliest recollections, when I began to study 
 physics out of the school-books in my father's posses- 
 sion, who was himself taught in the Cauer Institute, I 
 still see before me the dark image of a series of 
 ideas which seems now like the alchemy of the middle 
 ages. Of Lavoisier's and of Humphry Davy's revo- 
 lutionising discoveries, not much had got into the 
 school-books. Although oxygen was already known, 
 yet phlogiston, the fire element, played also its part. 
 Chlorine was still oxygenated hydrochloric acid ; potash 
 and lime were still elements. Invertebrate animals 
 were divided into insects and reptiles ; and in botany 
 we still counted stamens. 
 
 It is strange to see how late and with what hesita- 
 tion Germans applied themselves to the study of natural 
 
GUSTAV MAGNUS. 11 
 
 science in this century, whilst they had taken so promi- 
 nent a part in its earlier development. I need only 
 name Copernicus, Kepler, Leibnitz, and Stahl. 
 
 For we may indeed boast of our eager, fearless 
 and unselfish love of truth, which flinches before no 
 authority, and is stopped by no pretence ; shuns no 
 sacrifice and no labour, and is very modest in its claims 
 on worldly success. But even on this account she ever 
 impels us first of all to pursue the questions of prin- 
 ciple to their ultimate sources, and to trouble ourselves 
 but little about what has no connection with funda- 
 mental principles, and especially about practical con- 
 sequences and about useful applications. To this must 
 be added another reason, namely, that the independent 
 mental development of the last three hundred years, 
 began under political conditions which caused the 
 chief weight to fall on theological studies. Germany 
 has liberated Europe from the tyranny of the ancient 
 church ; but she has also paid a much dearer price for 
 this freedom than other nations. After the religious 
 wars, she remained devastated, impoverished, politi- 
 cally shattered, her boundaries spoiled, and arrogantly 
 handed over defenceless to her neighbours. To deduce 
 consequences from the new moral views, to prove them 
 scientifically, to work them out in all regions of intellec- 
 tual life, for all this there was no time during the storm 
 of war ; each man had to hold to his own party, every 
 
12 GUSTAV MAGNUS. 
 
 incipient change of opinion was looked upon as treach- 
 ery, and excited bitter wrath. Owing to the Keformation, 
 intellectual life had lost its old stability and cohesion ; 
 everything appeared in a new light, and new questions 
 arose. The Grerman 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 
 such 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 Grerman mind ; it could not again abandon it 
 until all its hiding-places had been searched through 
 and it had satisfied itself that for the present nothing 
 more is to be found there. 
 
GUSTAV MAGNUS. 13 
 
 Then, in the second half of the last century, the 
 rejuvenescent intellectual life of the nation began to 
 cultivate its artistic flowers ; the clumsy language trans- 
 formed itself into one of the most expressive instru- 
 ments of the human mind ; out of what was still the 
 hard, poor, and wearisome condition of civil and political 
 life, the results of the religious war, in which the figure 
 of the Prussian hero-king only now cast the first tope 
 of a better future, to be again followed by the misery 
 of the Napoleonic war, out of this joyless existence, 
 all sensitive minds gladly fled into the flowery land 
 opened out by German poetry, rivalling as it did the 
 best poetry of all times and of all peoples ; or in the 
 sublime aspects of philosophy they endeavoured to 
 sink reality in oblivion. 
 
 And the natural sciences were on the side of this 
 real world, so willingly overlooked. Astronomy alone 
 could at that time offer great and sublime prospects ; 
 in all other branches long and patient labour was still 
 necessary before great principles could be attained ; 
 before these subjects could have a voice in the great 
 questions of human life; or before they became the 
 powerful means of the authority of man over the 
 forces of nature which they have since become. The 
 labour of the natural philosopher seems narrow, low, 
 and insignificant compared with the great conceptions 
 of the philosopher and of the poet ; it was only those 
 
14 GUSTAV MAGNUS. 
 
 natural philosophers who, like Oken, rejoiced in 
 poetical philosophical conceptions, who found willing 
 auditors. 
 
 Far be it from me as a one-sided advocate of scien- 
 tific interests to blame this period of enthusiastic ex- 
 citement ; we have, in fact, to thank it for the moral 
 force which broke the Napoleonic yoke ; we have to 
 thank it for the grand poetry which is the noblest 
 treasure of our nation ; but the real world retains its 
 right against every semblance, even against the most 
 beautiful ; and individuals, as well as nations, who wish 
 to rise to the ripeness of manhood must learn to look 
 reality in the face, in order to bend it to the purpose of 
 the mind. To flee into an ideal world is a false re- 
 source of transient success ; it only facilitates the play 
 of the adversary ; and when knowledge only reflects 
 itself, it becomes unsubstantial and empty, or resolves 
 itself into illusions and phrases. 
 
 Against the errors of a mental tendency, which cor- 
 responded at first to the natural soaring of a fresh youth- 
 ful ambition, but which afterwards, in the age of the 
 Epigones of the Komantic school and of the philosophy 
 of Identity, fell into sentimental straining after sub- 
 limity and inspiration, a reaction took place, and was 
 carried out not merely in the regions of science, but 
 also in history, in art, and in philology. In the last 
 departments, too, where we deal directly with products 
 
GUSTAV MAGNUS. 15 
 
 of activity of the human mind, and where, therefore, 
 a construction a priori from the psychological laws 
 seems much more possible than in nature, it has come 
 to be understood that we must first know the facts, be- 
 fore we can establish their laws. 
 
 Gustav Magnus's development happened during the 
 period of this struggle ; it lay in the whole tendency 
 of his mind, that he whose gentle spirit usually en- 
 deavoured to reconcile antagonisms, took a decided part 
 in favour of pure experience as against speculation. 
 If he forbore to wound people, it must be confessed 
 that he did not relax one iota of the principle which, 
 with sure instinct, he had recognised as the true one ; 
 and in the most influential quarters he fought in a 
 twofold sense ; on the one hand, because in physics it 
 was a question as to the foundations of the whole of 
 natural sciences ; and on the other hand, because the 
 University of Berlin, with its numerous students, had 
 long been the stronghold of speculation. He con- 
 tinually preached to his pupils that no reasoning, 
 however plausible it might seem, avails against actual 
 fact, and that observation and experiment must de- 
 cide ; and he was always anxious that every practicable 
 experiment should be made which could give practical 
 confirmation or refutation of an assumed law. He did 
 not limit in any way the applicability of scientific 
 methods in the investigation of inanimate nature, but 
 
16 GUSTAV MAGNUS. 
 
 in his research on the gases of the blood (1837) he 
 dealt a blow at the heart of vitalistic theories. He led 
 physics to the centre of organic change, by laying a 
 scientific foundation for a correct theory of respira- 
 tion; a foundation upon which a great number of 
 more recent investigators have built, and which has 
 developed into one of the most important chapters 
 of physiology. 
 
 He cannot be reproached with having had too little 
 confidence in carrying out his principle; but I must 
 confess that I myself and many of my companions 
 formerly thought that Magnus carried his distrust of 
 speculation too far, especially in relation to mathe- 
 matical physics. He had probably never dipped very 
 deep in the latter subject, and that strengthened our 
 doubts. Yet when we look around us from the stand- 
 point which science has now attained, it must be con- 
 fessed that his distrust of the mathematical physics of 
 that date was not unfounded. At that time no separa- 
 tion had been distinctly made as to what was empirical 
 matter of fact, what mere verbal definition, and what 
 only hypothesis. The vague mixture of these ele- 
 ments which formed the basis of calculation was put 
 forth as axioms of metaphysical necessity, and pos- 
 tulated a similar kind of necessity for the results. I 
 need only recall to you the great part which hypo- 
 theses as to the atomic structure of bodies played 
 
GUSTAV MAGNUS. 17 
 
 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 Gauss, 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 
 
 c 
 
18 GUSTAV MAGNUS. 
 
 and constituted; only with such can we make our 
 observations and experiments. Their actions are made 
 up of the actions which each of their parts contributes 
 to the sum of the whole; and hence, if we wish to 
 know the simplest and most general law of the action 
 of masses and substances found in nature upon one 
 another, and if we wish to divest these laws of the 
 accidents of form, magnitude, and position of the 
 bodies concerned, we must go back to the laws of 
 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 of 
 the sum is sufficient to discover the constituents, 
 or hypothetically, where the calculated sum of effects 
 in the greatest possible number of different cases 
 must be compared with actual fact by observation 
 and by experiment. It is thus admitted that mathe- 
 matical physics only investigates the laws of action 
 of the elements of a body independently of the acci- 
 dents of form, in a purely empirical manner, and is there- 
 fore just as much under the control of experience as 
 what are called experimental physics. In principle 
 they are not at all different, and the former only con- 
 
GUSTAV MAGNUS. 19 
 
 tinues the function of the latter, in order to arrive at 
 still simpler and still more general laws of phenomena. 
 
 It cannot be doubted that this analytical tendency 
 of physical inquiry has assumed another character ; 
 that it has just cast off that which was the means of 
 placing Magnus towards it in some degree of antago- 
 nism. He tried to maintain, at least in former years, 
 that the business of the mathematical and that of 
 the experimental physicist are quite distinct from one 
 another; that a young man who wishes to pursue 
 physics would have to decide between the two. It 
 appears to me, on the contrary, that the conviction is 
 constantly gaining ground, that in the present more 
 advanced state of science those only can experi- 
 mentalise profitably who have a clear-sighted know- 
 ledge of theory, and know how to propound and pursue 
 the right questions ; and, on the other hand, only 
 those can theorise with advantage who have great 
 practice in experiments. The discovery of spectrum 
 analysis is the most brilliant example within our 
 recollection of such an interpenet ration 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 
 
 c 2 
 
20 GUSTAV MAGNUS. 
 
 too far, must also admit that in the older mathema- 
 tical physics there are many reasons for this dislike, 
 and that, on the other hand, he received with the 
 greatest pleasure the results which Kirchhoff, Sir 
 W. Thomson, and others had developed out of new 
 facts from theoretical starting-points. I may here be 
 permitted to adduce my own experience. My re- 
 searches were mostly developed in a manner against 
 which Magnus tried to guard; yet I never found in 
 him any but the most willing and friendly recognition. 
 It is, however, natural that every one, relying upon 
 his own experience, should recommend to others, as 
 most beneficial, the way which best suits his own 
 nature, and by which he has made the quickest pro- 
 gress. And if we are all of the same opinion that the 
 task of science is to find the Laws of Facts, then 
 each one may be left free either to plunge into facts, 
 and to search where he might come upon traces of 
 laws still unknown, or from laws already known to 
 search out the points where new facts are to be dis- 
 covered. But just as we all, like Magnus, are op- 
 posed to the theorist who holds it unnecessary to 
 prove experimentally the hypothetical results which 
 seem axioms to him, so would Magnus as his works 
 decidedly show pronounce with us against that kind 
 of excessive empiricism which sets out to discover 
 facts which fit to no rule, and which also try carefully 
 
GUSTAV MAGNUS. ?| 
 
 0* J 
 
 to avoid a law, or a possible connection between Jaewly 
 discovered facts. 
 
 It must here be mentioned that Faraday, another 
 great physicist, worked in England exactly in the 
 same direction, and with the same object ; to whom, 
 on that account, Magnus was bound by the heartiest 
 sympathy. With Faraday, the antagonism to the phy- 
 sical theories hitherto held, which treated of atoms 
 and forces acting at a distance, was even more pro- 
 nounced than with Magnus. 
 
 We must, moreover, admit that Magnus mostly 
 worked with success on problems which seemed 
 specially adapted to mathematical treatment; as, for 
 instance, his research on the deviation of rotating shot 
 fired from rifled guns ; also his paper on the form of 
 jets of water and their resolution into drops. In the 
 first, he proved, by a very cleverly arranged experi- 
 ment, how the resistance of the air, acting on the ball 
 from below, must deflect it sideways as a rotating 
 body, in a direction depending on that of the rotation ; 
 and how, in consequence of this, the trajectory is de- 
 flected in the same direction. In the second treatise, 
 he investigated the different forms of jets of water, 
 how they are partly changed by the form of the aper- 
 ture through which they flow, partly in consequence 
 of the manner in which they flow to it ; and how their 
 resolution into drops is caused by external agitation. 
 
22 GUSTAV MAGNUS. 
 
 He applied the principle of the stroboscopic disk in 
 observing the phenomena, by looking at the jet 
 through small slits in a rotating disk. He grouped 
 the various phenomena with peculiar tact, so that 
 those among them which are alike were easily seen, 
 and one elucidated the other. And if a final mechanical 
 explanation is not always attained, yet the reason for 
 a great number of characteristic features of the indi- 
 vidual phenomena is plain. In this respect many of 
 his researches I might specially commend those on 
 the efflux of jets of water are excellent models of 
 what Groethe theoretically advanced, and in his phy- 
 sical labours endeavoured to accomplish, though with 
 only partial success. 
 
 But even where Magnus from his standpoint, and 
 armed with the knowledge of his time, exerted himself 
 in vain to seize the 'kernel of the solution of a difficult 
 question, a host of new and valuable facts is always 
 brought to light. Thus in his research on the thermo- 
 electric battery, where he correctly saw that a critical 
 question was to be solved, and at the conclusion de- 
 clared : ' When I commenced the experiment just 
 described, I confidently hoped to find that thermo- 
 electrical currents are due to a motion of heat.' In 
 this sense he investigated the cases in which the 
 thermo-electrical circuit consisted of a single metal in 
 which there were alternately hard portions, and such as 
 
GUSTAV MAGNUS. 23 
 
 had been softened by heat ; or those in which the 
 parts in contact had very different temperatures. He 
 was convinced that the thermo-electrical current was 
 due neither to the radiating power, nor to the conduc- 
 tivity for heat, using this expression in its ordinary 
 meaning, and he had to content himself with the ob- 
 viously imperfect explanation that two pieces of the 
 same metal at different temperatures acted like dis- 
 similar conductors, which like liquids do not fall in with 
 the potential series. The solution was first furnished 
 by the two general laws of the mechanical theory of heat. 
 Magnus's hope was not unfulfilled. Sir W. Thomson 
 discovered that alterations in the conductivity for heat, 
 though such as were produced by the electrical current 
 itself, were indeed the sources of the current. 
 
 It is the nature of the scientific direction which 
 Magnus pursued in his researches, that they build 
 many a stone into the great fabric of science, which 
 give it an ever broader support, and an ever growing 
 height, without its appearing to a fresh observer as a 
 special and distinctive work due to the sole exertion 
 of any one scientific man. If we wish to explain the 
 importance of each stone for its special place, how 
 difficult to procure it, and how skilfully it was worked, 
 we must presuppose either that the hearer knows the 
 entire history of the building, or we must explain it to 
 him, by which more time is lost than I can now claim. 
 
24 GUSTAV MAGNUS. 
 
 Thus it is with Magnus's researches. Wherever he 
 has attacked, he has brought out a host of new and 
 often remarkable facts ; he has carefully and accurately 
 observed them, and he has brought them in connection 
 with the great fabric of science. He has, moreover, 
 bequeathed to science a great number of ingenious and 
 carefully devised new methods, as instruments with 
 which future generations will continue to discover 
 hidden veins of the noble metal of everlasting laws in 
 the apparently waste and wild chaos of accident. 
 Magnus's name will always be mentioned in the first 
 line of those on whose labours the proud edifice of the 
 science of Nature reposes ; of the science which has so 
 thoroughly remodelled the life of modern humanity by 
 its intellectual influence, as well as by its having subju- 
 gated the forces of nature to the dominion of the mind. 
 
 I have only spoken of Magnus's physical labours, 
 and have only mentioned those which seemed to me 
 characteristic for his individuality. But the number 
 of his researches is very great, and they extend over 
 wider regions than could now be grasped by any single 
 inquirer. He began as a chemist, but even then he 
 inclined to those cases which showed remarkable phy- 
 sical conditions ; he was afterwards exclusively a 
 physicist. But parallel with this he cultivated a very 
 extended study of technology, which of itself would 
 alone have occupied a man's life. 
 
GUSTAV MAGNUS. 25 
 
 He has departed, after a rich life and a fruitful 
 activity. The old law that no man's life is free from 
 pain must have been applied to him also ; and yet his 
 life seems to have been especially happy. He had 
 what men generally desire most ; but he knew how to 
 ennoble external fortune by putting it at the service of 
 unselfish objects. To him was granted, what is dearest 
 to the 'mind of a noble spirit, to dwell in the centre of 
 an affectionate family, and in a circle of faithful and 
 distinguished friends. But I count his rarest happi- 
 ness to be that he could work in pure enthusiasm for 
 an ideal principle ; and that he saw the cause which 
 he served go on victoriously, and develop to unheard 
 of wealth and ever wider activity. 
 
 And in conclusion we must add, in so far as 
 thoughtfulness, purity of intention, moral and intellec- 
 tual tact, modesty, and true humanity can rule over the 
 caprices of fortune and of man, in so far was Magnus 
 the artificer of his own fortune ; one of the most satis- 
 factory and contented natures, who secure the love 
 and favour of men, who with sure inspiration know 
 how to find the right place for their activity ; and of 
 whom we may say, envious fact does not embitter their 
 successes, for, working for pure objects and with pure 
 wishes, they would find contentment even without 
 external successes. 
 
ON THE 
 
 OEIGIN AND SIGNIFICANCE 
 
 OF 
 
 GEOMETEICAL AXIOMS. 
 
 Lecture delivered in the Docenten Verein in Heidelberg, 
 in the year 1870. 
 
 THE fact that a science can exist and can be de- 
 veloped as has been the case with geometry, has 
 always attracted the closest attention among those 
 who are interested in questions relating to the bases of 
 the theory of cognition. Of all branches of human 
 knowledge, there is none which, like it, has sprung as 
 a completely armed Minerva from the head of Jupiter ; 
 none before whose death-dealing Aegis doubt and in- 
 consistency have so little dared to raise their eyes. It 
 escapes the tedious and troublesome task of collect- 
 ing experimental facts, which is the province of the 
 natural sciences in the strict sense of the word ; the 
 
28 OK1GIN AND SIGNIFICANCE OF 
 
 sole form of its scientific method is deduction. Con- 
 clusion is deduced from conclusion, and yet no one 
 of common sense doubts but that these geometrical 
 principles must find their practical application in the 
 real world about us. Land surveying, as well as ar- 
 chitecture, the construction of machinery no less than 
 mathematical physics, are continually calculating re- 
 lations of space of the most varied kind by geometrical 
 principles ; they expect that the success of their con- 
 structions and experiments shall agree with these 
 calculations ; and no case is known in which this ex- 
 pectation has been falsified, provided the calculations 
 were made correctly and with sufficient data. 
 
 Indeed, the fact that geometry exists, and is cap- 
 able of all this, has always been used as a prominent 
 example in the discussion on that question, which 
 forms, as it were, the centre of all antitheses of philo- 
 sophical systems, that there can be a cognition of 
 principles destitute of any bases drawn from ex- 
 perience. In the answer to Kant's celebrated ques- 
 tion, c How are synthetical principles a priori 
 possible?' geometrical axioms are certainly those 
 examples which appear to show most decisively that 
 synthetical principles are a priori possible at all. 
 The circumstance that such principles exist, and force 
 themselves on our conviction, is regarded as a proof 
 that space is an a priori mode of all external perception. 
 
GEOMETRICAL AXIOMS. 29 
 
 It 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 the 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 (Ilrid. part ii. sect. i. * On the Axioms 
 of Intuition '), as axioms which express a priori the conditions of 
 intuition by the senses. It is not here the question, whether these 
 axioms were originally given as intuition of space, or whether they 
 are only the starting-points from which the understanding can 
 develop such axioms a priori on which my critic insists. 
 
30 ORIGIN AND SIGNIFICANCE OF 
 
 relations to experience, with the question whether it 
 is logically possible to replace them by others. 
 
 Seeing that the researches in question are more 
 immediately designed to furnish proofs for experts in 
 a region which, more than almost any other, requires 
 a higher power of abstraction, and that they are vir- 
 tually inaccessible to the non-mathematician, I will 
 endeavour to explain to such a one the question at 
 issue. I need scarcely remark that my explanation 
 will give no proof of the correctness of the new views. 
 He who seeks this proof must take the trouble to 
 study the original researches. 
 
 Anyone who has entered the gates of the first ele- 
 mentary axioms of geometry, that is, the mathematical 
 doctrine of space, finds on his path that unbroken 
 chain of conclusions of which I just spoke, by which 
 the ever more varied and more complicated figures 
 are brought within the domain of law. But even in 
 their first elements certain principles are laid down, 
 with respect to which geometry confesses that she 
 cannot prove them, and can only assume that anyone 
 who understands the essence of these principles will 
 at once admit their correctness. These are the so- 
 called axioms. 
 
 For example, the proposition that if the shortest 
 line drawn between two points is called a straight line, 
 there can be only one such straight line. Again, it is 
 
GEOMETRICAL AXIOMS. 31 
 
 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, line? 
 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 vthat of a line a line or a surface, 
 by that of a surface a surface or a solid, but by the 
 movement of a solid a solid and nothing else is 
 described. 
 
 Now what is the origin of such propositions, un- 
 questionably true yet incapable of proof in a science 
 where everything else is reasoned conclusion ? Are 
 they inherited from the divine source of our reason 
 as the idealistic philosophers think, or is it only that 
 the ingenuity of mathematicians has hitherto not been 
 penetrating enough to find the proof? Every new 
 votary, coming with fresh zeal to geometry, naturally 
 
32 OKIGIN AND SIGNIFICANCE OF 
 
 strives to succeed where all before him have failed. 
 And it is quite right that each should make the trial 
 afresh ; for, as the question has hitherto stood, it is 
 only by the fruitlessness of one's own efforts that one 
 can be convinced of the impossibility of finding a 
 proof. Meanwhile solitary inquirers are always from 
 time to time appearing who become so deeply en- 
 tangled in complicated trains of reasoning that they 
 can no longer discover their mistakes and believe they 
 have solved the problem. The axiom of parallels 
 especially has called forth a great number of seeming 
 demonstrations. 
 
 The main difficulty in these inquiries is, and always 
 has been, the readiness with which results of everyday 
 experience become mixed up as apparent necessities of 
 thought with the logical processes, so long as Euclid's 
 method of constructive intuition is exclusively followed 
 in geometry. It is in particular extremely difficult, on 
 this method, to be quite sure that in the steps, pre- 
 scribed for the demonstration we have not involun- 
 tarily and unconsciously drawn in some most general 
 results of experience, which the power of executing 
 certain parts of the operation has already taught us 
 practically. In drawing any subsidiary line for the 
 sake of his demonstration, the well-trained geometer 
 always asks if it is possible to draw such a line. It is 
 well known that problems of construction play an essen- 
 
GEOMETRICAL AXIOMS. 33 
 
 tial part in the system of geometry. At first sight, 
 these appear to be practical operations, introduced for 
 the training of learners; but in reality they estab- 
 lish the existence of definite figures. .They show that 
 points, straight lines, or circles such as the problem re- 
 quires to be constructed are possible under all con- 
 ditions, or they determine any exceptions that there 
 may be. The point on which the investigations turn, 
 that we are about to consider, is essentially of this 
 nature. The foundation of all proof by Euclid's 
 method consists in establishing the congruence of 
 lines, angles, plane figures, solids, &c. To make the 
 congruence evident, the geometrical figures are sup- 
 posed to be applied to one another, of course without 
 changing their form and dimensions. That this is 
 in fact possible we have all experienced from our 
 earliest youth. But, if we proceed to build necessities 
 of thought upon this assumption of the free trans- 
 lation of fixed figures, with unchanged form, to every 
 part of space, we must see whether the assumption 
 does not involve some presupposition of which no 
 logical proof is given. We shall see later on that it 
 does indeed contain one of the most serious import. 
 But if so, every proof by congruence rests upon a fact 
 which is obtained from experience only. 
 
 I offer these remarks, at first only to show what 
 difficulties attend the complete analysis of the pre- 
 
 D 
 
34 ORIGIN AND SIGNIFICANCE OF 
 
 suppositions we make, in employing the common con- 
 structive method. We evade them when we apply, to 
 the investigation of principles,, the analytical method 
 of modern algebraical geometry. The whole process 
 of algebraical calculation is a purely logical operation ; 
 it can yield no relation between the quantities sub- 
 mitted to it that is not already contained in the equa- 
 tions which give occasion for its being applied. The 
 recent investigations in question have accordingly been 
 conducted almost exclusively by means of the purely 
 abstract methods of analytical geometry. 
 
 However, after discovering by the abstract method 
 what are the points in question, we shall best get a 
 distinct view of them by taking a region of narrower 
 limits than our own world of space. Let us, as we 
 logically may, suppose reasoning beings of only two 
 dimensions to live and move on the surface of some 
 solid body. We will assume that they have not the 
 power of perceiving anything outside this surface, but 
 that upon it they have perceptions similar to ours. If 
 such beings worked out a geometry, they would of 
 course assign only two dimensions to their space. 
 They would ascertain that a point in moving describes 
 a line, and that a line in moving describes a surface. 
 But they could as little represent to themselves what 
 further spatial construction would be generated by a 
 surface moving out of itself, as we can represent what 
 
GEOMETRICAL AXIOMS. 35 
 
 would be generated by a solid moving out of the space 
 we know. By the much-abused expression ' to re- 
 present 'or 'to be able to think how something 
 happens ' I understand and I do not see how any- 
 thing else can be understood by it without loss of all 
 meaning the power of imagining the whole series of 
 sensible impressions that would be had in such a case. 
 Now as no sensible impression is known relating to 
 such an unheard-of event, as the movement to a fourth 
 dimension would be to us, or as a movement to our 
 third dimension would be to the inhabitants of a 
 surface, such a ' representation ' is as impossible as 
 the ' representation ' of colours would be to one born 
 blind, if a description of them in general terms could 
 be given to him. 
 
 Our surface-beings would also be able to draw 
 shortest lines in their superficial space. These would 
 not necessarily be straight lines in our sense, but what 
 are technically called geodetic lines of the surface on 
 which they li ve ; lines such as are described by a tense 
 thread laid along the surface, and which can slide upon 
 it freely. I will henceforth speak of such lines as the 
 straightest lines of any particular surface or given 
 space, so as to bring out their analogy with the 
 straight line in a plane. I hope by this expression to 
 make the conception more easy for the apprehension 
 
 D 2 
 
36 OEIGIN AND SIGNIFICANCE OF 
 
 of my non-mathematical hearers without giving rise 
 to misconception. 
 
 Now if beings of this kind lived on an infinite 
 plane, their geometry would be exactly the same as 
 our planimetry. They would affirm that only one 
 straight line is possible between two points ; that 
 through a third point lying without this line only one 
 line can be drawn parallel to it ; that the ends of a 
 straight line never meet though it is produced to 
 infinity, and soon. 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 smaller part of it is the shortest line 
 
GEOMETRICAL AXIOMS. 37 
 
 between its ends. Thus the notion of the geodetic or 
 straightest line is not quite identical with that of the 
 shortest line. If the two given points are the ends of 
 a diameter of the sphere, every plane passing through 
 this diameter cuts semicircles, on the surface of the 
 sphere, all of which are shortest lines between the 
 ends ; in which case there is an equal number of 
 equal shortest lines between the given points. Ac- 
 cordingly, the axiom of there being only one shortest 
 line between two points would not hold without a 
 certain exception for the dwellers on a sphere. 
 
 Of parallel lines the sphere-dwellers would know 
 nothing. They would maintain that any two straightest 
 lines, sufficiently produced, must finally cut not in one 
 only but in two points. The sum of the angles of a 
 triangle would be always greater than two right angles, 
 increasing as the surface of the triangle grew greater. 
 They could thus have no conception of geometrical 
 similarity between greater and smaller figures of the 
 same kind, for with them a greater triangle, must have 
 different angles from a smaller one. Their space 
 would be unlimited, but would be found to be finite <or 
 at least represented as such.. 
 
 It is clear, then, that such beings must set up a 
 very different system of geometrical axioms from that 
 of the inhabitants of a plane, or from ours with our 
 space of three dimensions, though the logical powers 
 
38 ORIGIN AND SIGNIFICANCE OF 
 
 of all were the same ; nor are more examples neces- 
 sary to show that geometrical axioms must vary ac- 
 cording to the kind of space inhabited by beings 
 whose powers of reason are quite in conformity with 
 ours. But let us proceed still farther. 
 
 Let us think of reasoning beings existing on the 
 surface of an egg-shaped body. Shortest lines could 
 be drawn between three points of such a surface and 
 a triangle constructed. But if the attempt were made 
 to construct congruent triangles at different parts of 
 the surface, it would be found that two triangles, with 
 three pairs of equal sides, would not have their angles 
 equal. The sum of the angles of a triangle drawn at 
 the sharper pole of the body would depart farther from 
 two right angles than if the triangle were drawn at the 
 blunter pole or at the equator. Hence it appears that 
 not even such a simple figure as a triangle can be 
 moved on such a surface without change of form. It 
 would also be found that if circles of equal radii were 
 constructed at different parts of such a surface (the 
 length of the radii being always measured by shortest 
 lines along the surface) the periphery would be greater 
 at the blunter than at the sharper end. 
 
 We see accordingly that, if a surface admits of the 
 figures lying on it being freely moved without change 
 of any of their lines and angles as measured along it, 
 the property is a special one and does not belong to 
 
GEOMETRICAL AXIOMS. 39 
 
 every kind of surface. The condition under which a 
 surface possesses this important property was pointed 
 out by Gauss in his celebrated treatise on the cur- 
 vature of surfaces. 1 The ' measure of curvature,' as he 
 called it, i.e. the reciprocal of the product of the 
 greatest and least radii of curvature, must be every- 
 where equal over the whole extent of the surface. 
 
 Gauss showed at the same time that this measure 
 of curvature is not changed if the surface is bent with- 
 out distension or contraction of any part of it. Thus 
 we can roll up a flat sheet of paper into the form of 
 a cylinder, or of a cone, without any change in the 
 dimensions of the figures taken along the surface of 
 the sheet. Or the hemispherical fundus of a bladder 
 may be rolled into a spindle-shape without altering the 
 dimensions on the surface. Geometry on a plane will 
 therefore be the same as on a cylindrical surface ; only 
 in the latter case we must imagine that any number of 
 layers of this surface, like the layers of a rolled sheet 
 of paper, lie one upon another, and that after each 
 entire revolution round the cylinder a new layer is 
 reached different from the previous ones. 
 
 These observations are necessary to give the reader a 
 notion of a kind of surface the geometry of which is on 
 the whole similar to that of the plane, but in which 
 
 1 Gauss, Werke, Bd. IV. p. 215, first published in Commentatianes 
 Soc. Reg. Scientt. Gottengensis recentiorcs, vol. vi., 1828. 
 
40 ORIGIN AND SIGNIFICANCE OF 
 
 the axiom of parallels does not hold good. This is a 
 kind of curved surface which is, as it were, geometri- 
 cally the counterpart of a sphere, and which has there- 
 fore been called the pseudospherical surface by the 
 distinguished Italian mathematician E. Beltrami, who 
 has investigated its properties. 1 It is a saddle-shaped 
 surface of which only limited pieces or strips can be 
 connectedly represented in our space, but which may 
 yet be thought of as infinitely continued in all direc- 
 tions, since each piece lying at the limit of the part 
 constructed can be conceived as drawn back to the 
 middle of it and then continued. The piece displaced 
 must in the process change its flexure but not its 
 dimensions, just as happens with a sheet of paper 
 moved about a cone formed out of a plane rolled up. 
 Such a sheet fits the conical surface in every part, but 
 must be more bent near the vertex and cannot be so 
 moved over the vertex as to be at the same time 
 adapted to the existing cone and to its imaginary 
 continuation beyond. 
 
 Like the plane and the sphere, pseudospherical sur- 
 faces have their measure of curvature constant, so that 
 every piece of them can be exactly applied to every 
 
 1 Saggio di Interpretazio-ne delta Geometric^ Non-Euclidea, Napoli, 
 1 868. Teoria fondamentale degli Spazii di Cwrvativra costante, An- 
 nali di Matematica, Ser. II. Tom. II. pp. 232-55. Both have 
 been translated into French by J. Hoiiel, Annales Scientifiques de 
 VEcole JVormale, Tom V., 1869. 
 
GEOMETRICAL AXIOMS. 41 
 
 other piece, and therefore all figures constructed at 
 one place on the surface can be transferred to any 
 other place with perfect congruity of form, and perfect 
 equality of all dimensions lying in the surface itself. 
 The measure of curvature as laid down by Gauss, 
 which is positive for the sphere and zero for the plane, 
 would have a constant negative value for pseudo- 
 spherical surfaces, because the two principal curvatures 
 of a saddle-shaped surface have their concavity turned 
 opposite ways. 
 
 A strip of a pseudospherical surface may, for exam- 
 ple, be represented by the inner surface (turned towards 
 the axis) of a solid anchor-ring. If the plane figure 
 aabb (Fig. 1) is made to revolve on its axis of symme- 
 try AB, the two arcs ab will describe a pseudospherical 
 concave-convex surface like that of the ring. Above 
 and below, towards aa and 66, the surface will turn 
 outwards with ever-increasing flexure, till it becomes 
 perpendicular to the axis, and ends at the edge with one 
 curvature infinite. Or, again, half of a pseudospheri- 
 cal surface may be rolled up into the shape of a cham- 
 pagne-glass (Fig. 2), with tapering stem infinitely 
 prolonged. But the surface is always necessarily 
 bounded by a sharp edge beyond which it cannot be 
 directly continued. Only by supposing each single 
 piece of the edge cut loose and drawn along the surface 
 of the ring or glass, can it be brought to places of 
 
42 
 
 ORIGIN AND SIGNIFICANCE OF 
 
 different flexure, at which farther continuation of the 
 piece is possible. 
 
 In this way too the straightest 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 
 
 FIG. l. 
 
 FIG. 2. 
 
 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- 
 
GEOMETRICAL AXIOMS. 43 
 
 batehewsky, professor of mathematics at Kasan, 1 and 
 it was proved that this system could be carried out as 
 consistently as Euclid's. It agrees exactly with the 
 geometry of the pseudospherical surfaces worked out 
 recently by Beltrami. 
 
 Thus we see that in the geometry of two dimen- 
 sions a surface is marked out as a plane, or a sphere, or 
 a pseudospherical surface, by the assumption that any 
 figure may be moved about in all directions without 
 change of dimensions. The axiom, that there is only 
 one shortest line between any two points, distinguishes 
 the plane and the pseudospherical surface from the 
 sphere, and the axiom of parallels marks off the plane 
 from the pseudosphere. These three axioms are in 
 fact necessary and sufficient, to define as a plane the 
 surface to which Euclid's planimetry has reference, as 
 distinguished from all other modes of space in two 
 dimensions. 
 
 The difference between plane and spherical geome- 
 try has been long evident, but the meaning of the 
 axiom of parallels could not be understood till Gauss 
 had developed the notion of surfaces flexible without 
 dilatation, and consequently that of the possibly in- 
 finite continuation of pseudospherical surfaces. In- 
 habiting, as we do, a space of three dimensions and 
 endowed with organs of sense for their perception, we 
 1 Principien der Geometric, Kasan, 1829-30. 
 
44 OEIGIN AND SIGNIFICANCE OF 
 
 can represent to ourselves the various cases in which 
 beings on a surface might have to develop their per- 
 ception of space ; for we have only to limit our own 
 perceptions to a narrower field. It is easy to think 
 away perceptions that we have ; but it is very difficult 
 to imagine perceptions to which there is nothing ana- 
 logous in our experience. When, therefore, we pass to 
 space of three dimensions, we are stopped in our power 
 of representation, by the structure of our organs and 
 the experiences got through them which correspond 
 only to the space in which we live. 
 
 There is however another way of treating geometry 
 scientifically. All known space-relations are measur- 
 able, that is, they may be brought to determination of 
 magnitudes (lines, angles, surfaces, volumes). Problems 
 in geometry can therefore be solved, by finding methods 
 of calculation for arriving at unknown magnitudes from 
 known ones. This is done in analytical geometry, where 
 all forms of space are treated only as quantities and 
 determined by means of other quantities. Even the 
 axioms themselves make reference to magnitudes. The 
 straight line is defined as the -shortest between two 
 points, which is a determination of quantity. The 
 axiom of parallels declares that if two straight lines in 
 a plane do not intersect (are parallel), the alternate 
 angles, or the corresponding angles, made by a third 
 line intersecting them, are equal; or it may be laid 
 
GEOMETEICAL AXIOMS. 45 
 
 down instead that the sum of the angles of any 
 triangle is equal to two right angles. These, also, 
 are determinations of quantity. 
 
 Now we may start with this view of space, accord- 
 ing to which the position of a point may be deter- 
 mined by measurements in relation to any given 
 figure (system of co-ordinates), taken as fixed, and 
 then inquire what are the special characteristics of our 
 space as manifested in the measurements that have 
 to be made, and how it differs from other extended 
 quantities of like variety. This path was first entered 
 by one too early lost to science, B. Eiemann of Gfott- 
 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. (Abhandl. der TtonigL 
 GeselhcJi. zu Gottingen, Bd. XIII.) 
 
46 OKIGIN AND SIGNIFICANCE OF 
 
 calls a system of differences in which one thing can be 
 determined by n measurements an '?ifold 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 admeasure 
 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. v Eiemann showed, 
 by considerations of this kind, that the essential foun- 
 
 1 Helmholtz's Popular Lectures, Series I. p. 243. 2 Ibid. p. 86. 
 
GEOMETRICAL AXIOMS. 47 
 
 dation of any system of geometry, is the expression 
 that it gives for the distance between two points lying 
 in any direction towards one another, beginning with 
 the infinitesimal interval. He took from analytical 
 geometry the most general form for this expression, 
 that, namely, which leaves altogether open the kind of 
 measurements by which the position of any point is 
 given. 1 Then he showed that the kind of free mobi- 
 lity without change of form which belongs to bodies 
 in our space can only exist when certain quantities 
 yielded by the calculation 2 quantities that coincide 
 with Gauss's measure of surface-curvature when they 
 are expressed for surfaces have everywhere an equal 
 value. For this reason Biemann 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. 
 
 3 As occurs, for instance, in the above-mentioned work of Tobias, 
 pp. 70, etc. 
 
48 OKIGIN AND SIGNIFICANCE OF 
 
 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 
 called 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 L 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 Teoriafondamentale, $e., ut snjj. 
 
GEOMETRICAL AXIOMS. 49 
 
 portrayed in the interior of a sphere in Euclid's homa- 
 loid space, that every straightest line. or flattest surface 
 of the pseudospherical space is represented by a 
 straight line or a plane, respectively, in the sphere. 
 The surface itself of the sphere corresponds to the 
 infinitely distant points of the pseudospherical space ; 
 and the different parts of this space, as represented in 
 the sphere, become smaller, the nearer they lie to the 
 spherical surface, diminishing more rapidly in the direc- 
 tion of the radii than in that perpendicular to them. 
 Straight lines in the sphere, which only intersect 
 beyond its surface, correspond to straightest lines of 
 the pseudospherical space which never intersect. 
 
 Thus it appeared that space, considered as a region 
 of measurable quantities, does not at all correspond 
 with the most general conception of an aggregate of 
 three dimensions, but involves also special conditions, 
 depending on the perfectly free mobility of solid 
 bodies without change of form to all parts of it and 
 with all possible changes of direction ; and, further, on 
 the special value of the measure of curvature which 
 for our actual space equals, or at least is not distin- 
 guishable from, zero. This latter definition is given 
 in the axioms of straight lines and parallels. 
 
 Whilst Kiemann entered upon this new field from 
 the side of the most general and fundamental questions 
 of analytical geometry, I myself arrived at similar 
 
 E 
 
50 OKIGIN AND SIGNIFICANCE OF 
 
 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. Biemann 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 Riemann 
 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 dirnen- 
 
 1 Ueber die Thatsachen die der Geometric zum Grunde liegen 
 (Nachrichtenvonder konigl. Ges. d. Wiss.zu Gottingen, Juni 3, 1868). 
 
GEO]fflETBICAL AXIOMS. 51 
 
 sions of the space in question. It is further assumed 
 that with the movement of the point A, the magnitudes 
 used as co-ordinates vary continuously. 
 
 Secondly, the definition of a solid body, or rigid 
 system of points, must be made in such a way as to 
 admit of magnitudes being compared by congruence. 
 As we must not, at this stage, assume any special 
 methods for the measurement of magnitudes, our defi- 
 nition can, in the first instance, run only as follows : 
 Between the co-ordinates of any two points belonging 
 to a sojid 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, 
 BC, BD, BE, 
 CD, CE, 
 DE, 
 
 B 2 
 
52 OKIGIN AND SIGNIFICANCE OF 
 
 and therefore ten equations, involving in space of three 
 dimensions fifteen variable co-ordinates. But of these 
 fifteen, six must remain arbitrary, if the system of five 
 points is to admit of free movement and rotation, and 
 thus the ten equations can determine only nine co-ordi- 
 nates as functions of the six variables. With six points 
 we obtain fifteen equations for twelve quantities, with 
 seven points twenty-one equations for fifteen, and so 
 on. Now from n independent equations we can 
 determine n contained quantities, and if we have 
 more than n equations, the superfluous ones must be 
 deducible from the first n. Hence it follows that the 
 equations which subsist between the co-ordinates of 
 each pair of points of a solid body must have a special 
 character, seeing that, when in space of three dimen- 
 sions they are satisfied for nine pairs of points as 
 formed out of any five points, the equation for the tenth 
 pair follows by logical consequence. Thus our assump- 
 tion for the definition of solidity, becomes quite suffi- 
 cient to determine the kind of equations holding be- 
 tween the co-ordinates of two points rigidly connected. 
 Thirdly, the calculation must further be based on 
 the fact of a peculiar circumstance in the movement of 
 solid bodies, a fact so familiar to us that but for this 
 inquiry it might never have been thought of as some- 
 thing that need not be. When in our space of three 
 dimensions two points of a solid body are kept fixed, 
 
GEOMETRICAL AXIOMS. 53 
 
 its movements are limited to rotations round the 
 straight line connecting them. If we turn it com- 
 pletely round once, it again occupies exactly the po- 
 sition it had at first. This fact, that rotation in one 
 direction always brings a solid body back into its ori- 
 ginal position, needs special mention. A system of 
 geometry is possible without it. This is most easily 
 seen in the geometry of a plane. Suppose that with 
 every rotation of a plane figure its linear dimensions in- 
 creased in proportion to the angle of rotation, the figure 
 after one whole rotation through 360 degrees would no 
 longer coincide with itself as it was originally. But 
 any second figure that was congruent with the first in 
 its original position might be made to coincide with it 
 in its second position by being also turned through 
 360 degrees. A consistent system of geometry would 
 be possible upon this supposition, , which does not come 
 under Riemann's formula. 
 
 On the other hand I have shown that the three 
 assumptions taken together form a sufficient basis for 
 the starting-point of Riemann's investigation,, and 
 thence for all his further- results relating to the dis- 
 tinction of different spaces according to their measure 
 of curvature. 
 
 It still remained to be seen whether the laws of 
 motion, as dependent on moving forces, could also 'be 
 consistently transferred to spherical or pseudospherical 
 
54 OKIGIN AND SIGNIFICANCE OF 
 
 space. This investigation has been carried out by 
 Professor Lipschitz of Bonn. 1 It is found that the 
 comprehensive expression for all the laws of dynamics, 
 Hamilton's principle, may be directly transferred to 
 spaces of which the measure of curvature is other than 
 zero. Accordingly, in this respect also, the disparate 
 systems of geometry lead to no contradiction. 
 
 We have now to seek an explanation of the special 
 characteristics of our own flat space, since it appears 
 that they are not implied in the general notion of an 
 extended quantity of three dimensions and of the free 
 mobility of bounded figures therein. Necessities of 
 thought, such as are involved in the conception of such 
 a variety, and its measurability, or from the most 
 general of all ideas of a solid figure contained in it, 
 and of its free mobility, they undoubtedly are not. 
 Let us then examine the opposite assumption as to 
 their origin being empirical, and see if they can be 
 inferred from facts of experience and so established, or 
 if, when tested by experience, they are perhaps to be 
 rejected. If they are of empirical origin, we must be 
 able to represent to ourselves connected series of facts, 
 indicating a different value for the measure of curva- 
 ture from that of Euclid's flat space. But if we can 
 
 1 ' Untersuchungen iiber die ganzen homogenen Functionen von n 
 Differentialen' (Borchardt's Journal fur Mathematik, Bd. Ixx. 3, 71 ; 
 Ixxiii. 3, 1) ; Untersuchung eines Problems der Variationsrechnung' 
 (Ibid. Bd. Ixxiv.). 
 
GEOMETEICAL AXIOMS. 55 
 
 imagine such spaces of other sorts, it cannot be main- 
 tained that the axioms of geometry are necessary con- 
 sequences of an a priori transcendental form of intui- 
 tion, as Kant thought. 
 
 The distinction between spherical, pseudospherical, 
 and Euclid's geometry depends, as was above observed, 
 on the value of a certain constant called, by Kiemann, 
 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 Eiemann observed, 
 whether the result might not be different if we could 
 use other than our limited base-lines, the greatest of 
 which is the major axis of the earth's orbit. 
 
56 OBIGIN AND SIGNIFICANCE OP 
 
 Meanwhile, we must not forget that all geometrical 
 measurements rest ultimately upon the principle of 
 congruence. We measure the distance between points 
 by applying to them the compass, rule, or chain. We 
 measure angles by bringing the divided circle or theo- 
 dolite to the vertex of the angle. We also determine 
 straight lines by the path of rays of light which in 
 our experience is rectilinear ; but that light travels in 
 shortest lines as long as it continues in a medium of 
 constant refraction would be equally true in space of a 
 different measure of curvature. Thus all our geo- 
 metrical measurements depend on our instruments 
 being really, as we consider them, invariable in form, 
 or at least on their undergoing no other than the small 
 changes we know of, as arising from variation of tem- 
 perature, or from gravity acting differently at different 
 places. 
 
 In measuring, we only employ the best and surest 
 means we know of to determine, what we otherwise are 
 in the habit of making out by sight and touch or by 
 pacing. Here our own body with its organs is the 
 instrument we carry about in space. Now it is the 
 hand, now the leg, that serves for a compass, or the eye 
 turning in all directions is our theodolite for measur- 
 ing arcs and angles in the visual field. 
 
 Every comparative estimate of magnitudes or mea- 
 surement of their spatial relations proceeds therefore 
 
GEOMETRICAL AXIOMS. 57 
 
 upon a supposition as to the behaviour of certain phy- 
 sical things, either the human body or other instru- 
 ments employed. The supposition may be in the 
 highest degree probable and in closest harmony with 
 all other physical relations known to us, but yet it 
 passes beyond the scope of pure space-intuition. 
 
 It is in fact possible to imagine conditions for 
 bodies apparently solid such that the measurements in 
 Euclid's space become what they would be in spherical 
 or pseudospherical space. Let me first remind the 
 reader that if all the linear dimensions of other bodies, 
 and our own, at the same time were diminished or in- 
 creased in like proportion, as for instance to half or 
 double their size, we should with our means of space- 
 perception be utterly unaware of the change. This 
 would also be the case if the distension or contraction 
 were different in different directions, provided that 
 our own body changed in the same manner, and further 
 that a body in rotating assumed at every moment, 
 without suffering or exerting mechanical resistance, 
 the amount of dilatation in its different dimensions 
 corresponding to its position at the time. Think of 
 the image of the world in a convex mirror. The 
 common silvered globes set up in gardens give the 
 essential features, only distorted by some optical ir- 
 regularities. A well-made convex mirror of moderate 
 aperture represents the objects in front of it as ap- 
 
58 OKIOIN AND SIGNIFICANCE OF 
 
 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 
 
GEOMETEICAL AXIOMS. 59 
 
 world as we can look into theirs, without overstepping 
 the boundary, they must declare it to be a picture in a 
 spherical mirror, and would speak of us just as we 
 speak of them ; and if two inhabitants of the different 
 worlds could communicate with one another, neither, 
 so far as I can see, would be able to convince the other 
 that he had the true, the other the distorted, relations. 
 Indeed I cannot see that such a question would have 
 any meaning at all, so long as mechanical considerations 
 are not mixed up with it. 
 
 Now Beltrami's representation of pseudospherical 
 space in a sphere of Euclid's space, is quite similar, ex- 
 cept that the background is not a plane as in the 
 convex mirror, but the surface of a sphere, and that 
 the proportion in which the images as they approach 
 the spherical surface contract, has a different mathe- 
 matical expression. 1 If we imagine then, conversely, 
 that in the sphere, for the interior of which Euclid's 
 axioms hold good, moving bodies contract as they 
 depart from the centre like the images in a convex 
 mirror, and in such a way that their representatives 
 in pseudospherical space retain their dimensions 
 unchanged, observers whose bodies were regularly 
 subjected to the same change would obtain the 
 same results from the geometrical measurements 
 they could make as if they lived in pseudospherical 
 space. 
 
 1 Compare the Appendix at the end of this Lecture. 
 
60 OKIGIN AND SIGNIFICANCE OF 
 
 We can even go a step further, and infer how the 
 objects in a pseudospherical world, were it possible to 
 enter one, would appear to an observer, whose eye- 
 measure and experiences of space had been gained like 
 ours in Euclid's space. Such an observer would con- 
 tinue to look upon rays of light or the lines of vision 
 as straight lines, such as are met with in flat space, 
 and as they really are in the spherical representation 
 of pseudospherical space. The visual image of the 
 objects in pseudospherical space would thus make the 
 same impression upon him as if he were at the centre 
 of Beltrami's sphere. He would think he saw the 
 most remote objects round about him at a finite 
 distance, 1 let us suppose a hundred feet off. But as 
 he approached these distant objects, they would dilate 
 before him, though more in the third dimension than 
 superficially, while behind him they would contract. 
 He would know that his eye judged wrongly. If he 
 saw two straight lines which in his estimate ran 
 parallel for the hundred feet to his world's end, he 
 would find on following them that the farther he 
 advanced the more they diverged, because of the 
 dilatation of all the objects to which he approached. 
 On the other hand, behind him, their distance would 
 seem to diminish, so that as he advanced they would 
 
 1 The reciprocal of the square of this distance, expressed in 
 negative quantity, would be the measure of curvature of the pseudo- 
 spherical space. 
 
GEOMETRICAL AXIOMS. 61 
 
 appear always to diverge more and more. But two 
 straight lines which from his first position seemed to 
 converge to one and the same point of the background 
 a hundred feet distant, would continue to do this 
 however far he went, and he would never reach their 
 point of intersection. 
 
 Now we can obtain exactly similar images of our 
 real world, if we look through a large convex lens of 
 corresponding negative focal length, or even through a 
 pair of convex spectacles if ground somewhat prisma- 
 tically to resemble pieces of one continuous larger lens. 
 With these, like the convex mirror, we see remote ob- 
 jects as if near to us, the most remote appearing no 
 farther distant than the focus of the lens. In going 
 about with this lens before the eyes, we find that the 
 objects we approach dilate exactly in the manner I 
 have described for pseudospherical space. Now any one 
 using a lens, were it even so strong as to have a focal 
 length of only sixty inches, to say nothing of a hun- 
 dred feet, would perhaps observe for the first moment 
 that he saw objects brought nearer. But after going 
 about a little the illusion would vanish, and in spite 
 of the false images he would judge of the distances 
 rightly. We have every reason to suppose that what 
 happens in a few hours to any one beginning to wear 
 spectacles would soon enough be experienced in pseu- 
 dospherical space. In short, pseudospherical space 
 
62 OKIGIN AND SIGNIFICANCE OF 
 
 would not seem to us very strange, comparatively 
 speaking; we should only at first be subject to illu- 
 sions in measuring by eye the size and distance of the 
 more remote objects. 
 
 There would be illusions of an opposite description, 
 if, with eyes practised to measure in Euclid's space, we 
 entered a spherical space of three dimensions. We 
 should suppose the more distant objects to be more 
 remote and larger than they are, and should find on 
 approaching them that we reached them more quickly 
 than we expected from their appearance. But we 
 should also see before us objects that we can fixate 
 only with diverging lines of sight, namely, all those 
 at a greater distance from us than the quadrant of a 
 great circle. Such an aspect of things would hardly 
 strike us as very extraordinary, for we can have it even 
 as things are if we place before the eye a slightly pris- 
 matic glass with the thicker side towards the nose : the 
 eyes must then become divergent to take in distant 
 objects. This excites a certain feeling of unwonted 
 strain in the eyes, but does not perceptibly change the 
 appearance of the objects thus seen. The strangest 
 sight, however, in the spherical world would be the 
 back of our own head, in which all visual lines not 
 stopped by other objects would meet again, and which 
 must fill the extreme background of the whole per- 
 spective picture. 
 
GEOMETRICAL AXIOMS. 63 
 
 At the same time it must be noted that as a small 
 elastic flat disk, say of india-rubber, can only be fitted 
 to a slightly curved spherical surface with relative con- 
 traction of its border and distension of its centre, so 
 our bodies, developed in Euclid's flat space, could not 
 pass into curved space without undergoing similar 
 distensions and contractions of their parts, their co- 
 herence being of course maintained only in as far as 
 their elasticity permitted their bending without break- 
 ing. The kind of distension must be the same as in 
 passing from a small body imagined at the centre of 
 Beltrami's sphere to its pseudospherical or spherical 
 representation. For such passage to appear possible, 
 it will always have to be assumed that the body is 
 sufficiently elastic and small in comparison with the 
 real or imaginary radius of curvature of the curved 
 space into which it is to pass. 
 
 These remarks will suffice to show the way in 
 which we can infer from the known laws of our sen- 
 sible perceptions the series of sensible impressions 
 which a spherical or pseudospherical world would give 
 us, if it existed. In doing so, we nowhere meet with 
 inconsistency or impossibility any more than in the 
 calculation of its metrical proportions. We can re- 
 present to ourselves the look of a pseudospherical 
 world in all directions just as we can develop the con- 
 ception of it. Therefore it cannot be allowed that the 
 
64 ORIGIN AND SIGNIFICANCE OF 
 
 axioms of our geometry depend on the native form of 
 our perceptive faculty, or are in any way connected 
 with it. 
 
 It is different with the three dimensions of space. 
 As all our means of sense-perception extend only to 
 space of three dimensions, and a fourth is not merely 
 a modification of what we have, but something per- 
 fectly new, we find ourselves by reason of our bodily 
 organisation quite unable to represent a fourth di- 
 mension. 
 
 In conclusion, I would again urge that the axioms 
 of geometry are not propositions pertaining only to 
 the pure doctrine of space. As I said before, they are 
 concerned with quantity. We can speak of quantities 
 only when we know of some way by which we can com- 
 pare, divide, and measure them. All space-measure- 
 ments, and therefore in general all ideas of quantities 
 applied to space, assume the possibility of figures mov- 
 ing without change of form or size. It is true we are 
 accustomed in geometry to call such figures purely 
 geometrical solids, surfaces, angles, and lines, because 
 we abstract from all the other distinctions, physical 
 and chemical, of natural bodies ; but yet one physical 
 quality, rigidity, is retained. Now we have no other 
 mark of rigidity of bodies or figures but congruence, 
 whenever they are applied to one another at any time 
 or place, and after any revolution. We cannot, how- 
 
GEOMETRICAL AXIOMS. 65 
 
 ever, decide by pure geometry, and without mechanical 
 considerations, whether the coinciding bodies may not 
 both have varied in the same sense. 
 
 If it were useful for any purpose, we might with 
 perfect consistency look upon the space in which we 
 live as the apparent space behind a convex mirror with 
 its shortened and contracted background ; or we might 
 consider a bounded sphere of our space, beyond the 
 limits of which we perceive nothing further, as infinite 
 pseudospherical space. Only then we should have to 
 ascribe to the bodies which appear to us to be solid, and 
 to our own body at the same time, corresponding disten- 
 sions and contractions, and we should have to change 
 our system of mechanical principles entirely ; for even 
 the proposition that every point in motion, if acted upon 
 by no force, continues to move with unchanged velo- 
 city in a straight line, is not adapted to the image of 
 the world in the convex-mirror. The path would in- 
 deed be straight, but the velocity would depend upon 
 the place. 
 
 Thus the axioms of geometry are not concerned 
 with space-relations only but also at the same time 
 with the mechanical deportment of solidest bodies in 
 motion. The notion of rigid geometrical figure might 
 indeed be conceived as transcendental in Kant's sense, 
 namely, as formed independently of actual experience, 
 which need not exactly correspond therewith, any more 
 
66 ORIGIN AND SIGNIFICANCE OF 
 
 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- 
 ic solids of great resistance, retain the same form in 
 
GEOMETRICAL AXIOMS. 67 
 
 every part of space if no external force affects them, 
 is a single case falling under the general principle. 
 
 In conclusion, I do not, of course, maintain that man- 
 kind first arrived at space-intuitions, in agreement with 
 the axioms of Euclid, by any carefully executed systems 
 of exact measurement. It was rather a succession of 
 everyday experiences, especially the perception of the 
 geometrical similarity of great and small bodies, only 
 possible in flat space, that led to the rejection, as im- 
 possible, of every geometrical representation at variance 
 with this fact. For this no knowledge of the neces- 
 sary logical connection between the observed fact of 
 geometrical similarity and the axioms was needed ; but 
 only an intuitive apprehension of the typical relations 
 between lines, planes, angles, &c., obtained by nume- 
 rous and attentive observations an intuition of the 
 kind the. artist possesses of the objects he is to repre- 
 sent, and by means of which he decides with certainty 
 and accuracy whether a new combination, which he tries, 
 will correspond or not with their nature. It is true 
 that we have no word but intuition to mark this ; but 
 it is knowledge empirically gained by the aggregation 
 and reinforcement of similar recurrent impressions in 
 memory, and not a transcendental form given before 
 experience. That other such empirical intuitions of 
 fixed typical relations, when not clearly comprehended, 
 have frequently enough been taken by metaphysicians 
 
68 ORIGIN AND SIGNIFICANCE OF 
 
 for a priori principles, is a point on which I need 
 not insist. 
 
 To sum up, the final outcome of the whole inquiry 
 may be thus expressed : 
 
 (1.) The axioms of geometry, taken by themselves 
 out of all connection with mechanical propositions, re- 
 present no relations of real things. When thus iso- 
 lated, if we regard them with Kant as forms of 
 intuition transcendentally given, they constitute a 
 form into which any empirical content whatever will 
 fit, and which therefore does not in any way limit or 
 determine beforehand the nature of the content. This 
 is true, however, not only of Euclid's axioms, but also 
 of the axioms of spherical and pseudospherical geo- 
 metry. 
 
 (2.) As soon as certain principles of mechanics are 
 conjoined with the axioms of geometry, we obtain a 
 system of propositions which has real import, and 
 which can be verified or overturned by empirical obser- 
 vations, just as it can be inferred from experience. If 
 such a system were to be taken as a transcendental 
 form of intuition and thought, there must be assumed 
 a pre-established harmony between form and reality. 
 
GEOMETEICAL AXIOMS. 69 
 
 APPENDIX. 
 
 THE elements of the geometry of spherical space are most 
 easily obtained by putting for space of four dimensions the 
 equation for the sphere 
 
 aja + y* + aa + s=JKa ........ (1.) 
 
 and for the distance ds between the points (x, y, z, t) and 
 [(x+dx) (y+dy) (z+dz) (t + dl)] 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 
 
 in which a, b, c,f, as well as a, /3, y, <, are constants. 
 
 The length of the shortest arc, , between the points 
 (x y y, z, t), and (, rj, , r) follows, as in the sphere, from the 
 equation 
 
 
 One of the co-ordinates may be eliminated from the values 
 given in 2 to 4, by means of equation 1, and the expressions 
 then apply to space of three dimensions. 
 
 If we take the distances from the points 
 
 =,,==0 
 
70 ORIGIN AND SIGNIFICANCE OF GEOMETRICAL AXIOMS. 
 
 from which equation 1 gives 7=R, then, 
 
 fs n \_<r 
 sm (j) M 
 
 in which o-=\/a; 2 + 2/ 2 + 2;2 
 
 or, s =fi . arc sin (jA=R . arc tang (jj- (5.) 
 
 In this, SQ is the distance of the point x, y, z, measured 
 from the centre of the co-ordinates. 
 
 If now we suppose the point x, y, z, of spherical space, 
 to be projected in a point of plane space whose co-ordinates 
 are respectively 
 
 _Rx _Ry^ ^fe 
 t t 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 lines of spherical space 
 are represented in the system of *-, JT, , by straight lines. 
 For very small values of x, y, z, t=R, and 
 
 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 
 
 Sn=.K . arc 
 
 tang ( ) - - - (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 < -| J??r, and one for which S Q > J RTT. The 
 extension in the direction of r is then 
 
APPENDIX. 71 
 
 In order to obtain corresponding expressions for pseudo- 
 spherical space, let R and t be imaginary; that is, 7?=JiT, 
 and =ti. Equation 6 gives then 
 
 tang -& = 
 
 1$ i$ 
 
 from which, eliminating the imaginary form, we get 
 
 Here s has real values only as long as r=R; for r=H the 
 distance s in pseudospherical space is infinite. The image 
 in plane space is, on the contrary, contained in the sphere of 
 radius R, and every point of this sphere forms only one 
 point of the infinite pseudospherical space. The extension 
 in the direction of r is 
 
 *o_ y 
 
 dc ~V-i 2 
 
 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 
 
LI B R A R 1 
 
 \I YKKSITY 01 
 
 CALIFORNIA. 
 
 - 
 
 osr 
 THE EELATION OF OPTICS 
 
 TO 
 
 PAINTING. 
 
 Being the substance of a series of Lectures delivered in 
 Cologne, Berlin, and Bonn. 
 
 I FEAR that the announcement of my intention to ad- 
 dress you on the subject of plastic art may have created 
 no little- surprise among some of my hearers. For I 
 cannot doubt that many of you have had more fre- 
 quent opportunities of viewing works of art, and have 
 more thoroughly studied its historical aspects, than I can 
 lay claim to have done ; or indeed have had personal 
 experience in the actual practice of art, in which I am 
 entirely wanting. I have arrived at my artistic studies 
 by a path which is but little trod, that is, by the phy- 
 siology of the senses ; and in reference to those who 
 have a long acquaintance with, and who are quite at 
 home in the beautiful fields of art, I may compare 
 
74 ON THE KELATION OF OPTICS TO PAINTING. 
 
 myself to a traveller who has entered upon them by 
 a steep and stony mountain path, but who, in doing 
 so, has passed many a stage from which a good point 
 of view is obtained. If therefore I relate to you what 
 I consider I have observed, it is with the understand- 
 ing that I wish to regard myself as open to instruction 
 by those more experienced than myself. 
 
 The physiological study of the manner in which 
 the perceptions of our senses originate, how impressions 
 from without pass into our nerves, and how the condi- 
 tion of the latter is thereby altered, presents many 
 points of contact with the theory of the fine arts. On 
 a former occasion I endeavoured to establish such a 
 relation between the physiology of the sense of hearing, 
 and the theory of music. Those relations in that case 
 are particularly clear and distinct, because the elemen- 
 tary forms of music depend more closely on the nature 
 and on the peculiarities of our perceptions than is the 
 case in other arts, in which the nature of the material 
 to be used and of the objects to be represented has 
 a far greater influence. Yet even in those other 
 branches of art, the especial mode of perception of 
 that organ of sense by which the impression is taken 
 up is not without importance ; and a theoretical in- 
 sight into its action, and into the principle of its 
 methods, cannot be complete if this physiological ele- 
 ment is not taken into account. Next to music this 
 
ON THE RELATION OF OPTICS TO PAINTING. 75 
 
 seems to predominate more particularly in painting, 
 and this is the reason why I have chosen painting as 
 the subject of my present lecture. 
 
 The more immediate object of the painter is to 
 produce in us by his palette a lively visual impression 
 of the objects which he has endeavoured to represent. 
 The aim, in a certain sense, is to produce a kind of 
 optical illusion ; not indeed that, like the birds who 
 pecked at the painted grapes of Apelles, we are to sup- 
 pose we have present the real objects themselves, and 
 not a picture ; but in so far that the artistic represen- 
 tation produces in us a conception of their objects as 
 vivid and as powerful as if we had them actually before 
 us. The study of what are called illusions of the senses 
 is however a very prominent and important part of 
 the physiology of the senses ; for just those cases in 
 which external impressions evoke conceptions which 
 are not in accordance with reality are particularly in- 
 structive for discovering the laws of those means and 
 processes by which normal perceptions originate. We 
 must look upon artists as persons whose observation 
 of sensuous impressions is particularly vivid and accu- 
 rate, and whose memory for these images is particu- 
 larly true. That which long tradition has handed 
 down to the men most gifted in this respect, and 
 that which they have found by innumerable experi- 
 ments in the most varied directions, as regards means, 
 
76 ON THE RELATION OF OPTICS TO PAINTING. 
 
 and methods of representation, forms a series of import- 
 ant and significant facts, which the physiologist, who 
 has here to learn from the artist, cannot afford to ne- 
 glect. The study of works of art will throw great light 
 on the question as to which elements and relations of 
 our visual impressions are most predominant in deter- 
 mining our conception of what is seen, and what others 
 are of less importance. As far as lies within his power, 
 the artist will seek to foster the former at the cost of 
 the latter. 
 
 In this sense then a careful observation of the 
 works of the great masters will be serviceable, not only 
 to physiological optics, but also because the investigation 
 of the laws of the perceptions and of the observations 
 of the senses will promote the theory of art, that is, 
 the comprehension of its mode of action. 
 
 We have not here to do with a discussion of the 
 ultimate objects and aims of art, but only with an ex- 
 amination of the action of the elementary means with 
 which it works. The knowledge of the latter must, 
 however, form an indispensable basis for the solution 
 of the deeper questions, if we are to understand the 
 problems which the artist has to solve, and the mode 
 in which he attempts to attain his object. 
 
 I need scarcely lay stress on the fact, following as 
 it does from what I have already said, that it is not 
 my intention to furnish instructions according to which 
 
ON THE EELATION OF OPTICS TO PAINTING. 77 
 
 the artist is to work. I consider it a mistake to sup- 
 pose that any kind of aesthetic lectures such as these 
 can ever do so ; but it is a mistake which those very 
 frequently make who have only practical objects in 
 view. 
 
78 ON THE KEIATION OF OPTICS TO PAINTING. 
 
 <u< 
 
 The painter seeks to produce in his picture an image 
 of external objects. The first aim of our investigation 
 must be to ascertain what degree and what kind of 
 similarity he can expect to attain, and what limits are 
 assigned to him by the nature of his method. The 
 uneducated observer usually requires nothing more 
 than an illusive resemblance to nature : the more this 
 is obtained, the more does he delight in the picture. 
 An observer, on the contrary, whose taste in works of 
 art has been more finely educated, will, consciously or 
 unconsciously, require something more, and something 
 different. A faithful copy of crude Nature he will at 
 most regard as an artistic feat. To satisfy him, he 
 will need artistic selection, grouping, and even idealisa- 
 tion of the objects represented. The human figures 
 in a work of art must not be the everyday figures, 
 such as we see in photographs ; they must have ex- 
 pression, and a characteristic development, and if 
 possible beautiful forms, which have perhaps be- 
 longed to no living individuals or indeed any indi- 
 viduals which ever have existed, but only to such a 
 one as might exist, and as must exist, to produce a 
 
ON THE RELATION OF OPTICS TO PAINTING. 79 
 
 vivid perception of any particular aspect of human 
 existence in its complete and unhindered development. 
 
 If however the artist is to produce an artistic 
 arrangement of only idealised types, whether of man 
 or of natural objects, must not the picture be an 
 actual, complete, and directly true delineation of that 
 which would appear if it anywhere came into being ? 
 
 Since the picture is on a plane surface, this faith- 
 ful representation can of course only give a faithful 
 perspective view of the objects. Yet our eye, which 
 in its optical properties is equivalent to a camera 
 obscura, the well-known apparatus of the photo- 
 grapher, gives on the retina, which is its sensitive 
 plate, only perspective views of the external world ; 
 these are stationary, like the drawing on a picture, 
 as long as the standpoint of the eye is not altered. 
 And, in fact, if we restrict ourselves in the first place 
 to the form of the object viewed, and disregard for 
 the present any consideration of colour, by a correct 
 perspective drawing we can present to the eye of an 
 observer, who views it from a correctly chosen point 
 of view, the same forms of the visual image as the 
 inspection of the objects themselves would present to 
 the same eye, when viewed from the corresponding 
 point of view. 
 
 But apart from the fact that any movement of the 
 observer, whereby his eye changes its position, will 
 
80 ON THE RELATION OF OPTICS TO PAINTING: 
 
 produce displacements of the visual image, different 
 when he stands before objects from those when he 
 stands before the image, I could speak of only one 
 eye for which equality of impression is to be estab- 
 lished. We however see the world with two eyes, 
 which occupy somewhat different positions in space, 
 and which therefore show two different perspective 
 views of objects before us. This difference of the 
 images of the two eyes forms one of the most im- 
 portant means of estimating the distance of objects 
 from our eye, and of estimating depth, and this is 
 what is wanting to the painter, or even turns against 
 him; since in binocular vision the picture distinctly 
 forces itself on our perception as a plane surface. 
 
 You must all have observed the wonderful vividness 
 which the solid form of objects acquires when good 
 stereoscopic images are viewed in the stereoscope, a 
 kind of vividness in which either of the pictures is 
 wanting when viewed without the stereoscope. The 
 illusion is most striking and instructive with figures in 
 simple line ; models of crystals and the like, in 
 which there is no other element of illusion. The 
 reason of this deception is, that looking with two eyes 
 we view the world simultaneously from somewhat 
 different points of view, and thereby acquire two dif- 
 ferent perspective images. With the right eye we see 
 somewhat more of the right side of objects before us, 
 
ON THE RELATION OF OPTICS TO PAINTING. 81 
 
 and also somewhat more of those behind it, than we 
 do with the left eye ; and conversely we see with the 
 left, more of the left side of an object, and of the back- 
 ground behind its left edges, and partially concealed 
 by the edge. But a flat picture shows to the right eye 
 absolutely the same picture, and all objects represented 
 upon it, as to the left eye. If then we make for each 
 eye such a picture as that eye would perceive if itself 
 looked at the object, and if both pictures are combined 
 in the stereoscope, so that each eye sees its correspond- 
 ing picture, then as far as form is concerned the 
 same impression is produced in the two eyes as the 
 object itself produces. But if we look at a drawing or 
 a picture with both eyes, we just as easily recognise 
 that it is a representation on a plane surface, which is 
 different from that which the actual object would show 
 simultaneously to both eyes. Hence is due the well- 
 known increase in the vividness of a picture if it is 
 looked at with only one eye, and while quite stationary, 
 through a dark tube ; we thus exclude any comparison 
 of its distance with that of adjacent objects in the 
 room. For it must be observed that as we use differ- 
 ent pictures seen with the two eyes for the perception 
 of depth, in like manner as the body moves from one 
 place to another, the pictures seen by the same eye 
 serve for the same purpose. In moving, whether on 
 foot or riding, the nearer objects are apparently dis- 
 
 G 
 
82 ON THE KELATION OF OPTICS TO PAINTING. 
 
 placed in comparison with the more distant ones ; the 
 former appear to recede, the latter appear to move with 
 us. Hence arises a far stricter distinction between what 
 is near and what is distant, than seeing with one eye 
 from one and the same spot would ever afford us. If 
 we move towards the picture, the sensuous impression 
 that it is a flat picture hanging against the wall forces 
 itself more strongly upon us than if we look at it while 
 we are stationary. Compared with a large picture at a 
 greater distance, all those elements which depend on bin- 
 ocular vision and on the movement of the body are less 
 operative, because in very distant objects the differ- 
 ences between the images of the two eyes, or be- 
 tween the aspect from adjacent points of view, seem 
 less. Hence large pictures furnish a less distorted 
 aspect of their object than small ones, while the 
 impression on a stationary eye, of a small picture close 
 at hand, might be just the same as that of a large 
 distant one. In a painting close at hand, the fact that 
 it is a flat picture continually forces itself more power- 
 fully and more distinctly on our perception. 
 
 The fact that perspective drawings, which are taken 
 from too near a point of view, may easily produce a 
 distorted impression, is, I think, connected with this. 
 For here the want of the second representation for the 
 other eye, which would be very different, is too marked. 
 On the other hand, what are called geometrical pro- 
 
OX THE KELATION OF OPTICS TO PAINTING-. 83 
 
 jections, that is, perspective drawings which represent 
 a view taken from an infinite distance, give in many 
 cases a particularly favourable view of the object, 
 although they correspond to a point of sight which 
 does not in reality occur. Here the pictures of both 
 eyes for such an object are the same. 
 
 You will notice that in these respects there is a 
 primary incongruity, and one which cannot be got 
 over, between the aspect of a picture and the aspect 
 of reality. This incongruity may be lessened, but 
 never entirely overcome. Owing to the imperfect 
 action of binocular vision, the most important natural 
 means is lost of enabling the observer to estimate 
 the depth of objects represented in the picture. The 
 painter possesses a series of subordinate means, partly 
 of limited applicability, and partly of slight effect, 
 of expressing various distances by depth. It is not 
 unimportant to become acquainted with these elements, 
 as arising out of theoretical considerations ; for in the 
 practice of the art of painting they have manifestly 
 exercised great influence on the arrangement, selec- 
 tion, and mode of illumination of the objects repre- 
 sented. The distinctness of what is represented is 
 indeed of subordinate importance when considered in 
 reference to the ideal aims of art ; it must not however 
 be depreciated, for it is the first condition by which 
 the observer attains an intelligibility of expres- 
 
 G 2 
 
84 ON THE KELATION OF OPTICS TO PAINTING. 
 
 sion, which impresses itself without fatigue on the 
 observer. 
 
 This direct intelligibility is again the preliminary 
 condition for an undisturbed, and vivid action of the 
 picture on the feelings and mood of the observer. 
 
 The subordinate methods of expressing depth which 
 have been referred to, depend in the first place on per- 
 spective. Nearer objects partially conceal more distant 
 ones, but can never themselves be concealed by the 
 latter. If therefore the painter skilfully groups his ob- 
 jects, so that the feature in question comes into play, 
 this gives at once a very certain gradation of far and 
 near. This mutual concealment may even preponderate 
 over the binocular perception of depth, if stereoscopic 
 pictures are intentionally produced in which each coun- 
 teracts the other. Moreover, in bodies of regular or of 
 known form, the forms of perspective projection are for 
 the most part characteristic for the depth of the object. 
 If we look at houses, or other results of man's artistic 
 activity, we know at the outset that the forms are for the 
 most part plane surfaces at right angles to each other, 
 with occasional circular or even spheroidal surfaces. And 
 in fact, when we know so much, a correct perspective 
 drawing is sufficient to produce the whole shape of the 
 body. This is also the case with the figures of men and 
 animals which are familiar to us, and whose forms 
 moreover show two symmetrical halves. The best per- 
 
ON THE RELATION OF OPTICS TO PAINTING. 85 
 
 spective drawing is however of but little avail in the 
 case of irregular shapes, rough blocks of rock and ice, 
 masses of foliage, and the like ; that this is so, is best 
 seen in photographs, where the perspective and shading 
 may be absolutely correct, and yet the total impression 
 is indistinct and confused. 
 
 When human habitations are seen in a picture, they 
 represent to the observer the direction of the hori- 
 zontal surfaces at the place at which they stand ; and 
 in comparison therewith the inclination of the ground, 
 which without them would often be difficult to repre- 
 sent. 
 
 The apparent magnitude which objects, whose 
 actual magnitude is known, present in different parts 
 of the picture must also be taken into account. Men 
 and animals, as well as familiar trees, are useful to the 
 painter in this respect. In the more distant centre of 
 the landscape they appear smaller than in the fore- 
 ground, and thus their apparent magnitude furnishes 
 a measure of the distance at which they are placed. 
 
 Shadows, and more especially double ones, are of 
 great importance. You all know how much more 
 distinct is the impression which a well-shaded drawing 
 gives as distinguished from an outline ; the shading is 
 hence one of the most difficult, but at the same time 
 most effective, elements in the productions of the 
 draughtsman and painter. It is his task to imitate 
 
86 ON THE KELATION OF OPTICS TO PAINTING. 
 
 the fine gradation and transitions of light and shade 
 on rounded surfaces, which are his chief means of ex- 
 pressing their modelling, with all their fine changes of 
 curvature ; he must take into account the extension or 
 restriction of the sources of light, and the mutual 
 reflection of the surfaces on each other. While the 
 modifications of the lighting on the surface of bodies 
 themselves is often dubious for instance, an intaglio 
 of a medal may, with a particular illumination, pro- 
 duce the impression of reliefs which are only illumi- 
 nated from the other side double shadows, on the 
 contrary, are undoubted indications that the body which 
 throws the shadow is nearer the source of light than 
 that which receives the shadow. This rule is so com- 
 pletely without exception, that even in stereoscopic 
 views a falsely placed double shadow may destroy or 
 confuse the entire illusion. 
 
 The various kinds of illumination are not all equally 
 favourable for obtaining the full effect of shadows. 
 When the observer looks at the objects in the same 
 direction as that in which light falls upon them, he 
 sees only their illuminated sides and nothing of the 
 shadow ; the whole relief which the shadows could give 
 then disappears. If the object is between the source 
 of light and the observer he only sees the shadows. 
 Hence we need lateral illumination for a picturesque 
 shading ; and over surfaces which like those of plane 
 
ON THE RELATION OF OPTICS TO PAINTING. 87 
 
 or hilly land only present slightly moving figures, we 
 require light which is almost in the direction of the 
 surface itself, for only such a one gives shadows. This 
 is one of the reasons which makes illumination by the 
 rising or the setting sun so effective. The forms of 
 the landscape become more distinct. To this must 
 also be added the influence of colour, and of aerial 
 light, which we shall subsequently discuss. 
 
 Direct illumination from the sun, or from a flame, 
 makes the shadows sharply defined, and hard. Illu- 
 mination from a very wide luminous surface, such as 
 a cloudy sky, makes them confused, or destroys them 
 altogether. Between these two extremes there are 
 transitions ; illumination by a portion of the sky, 
 defined by a window, or -by trees, &c., allows the 
 shadows to be more or less prominent according to 
 the nature of the object. You must have seen of 
 what importance this is to photographers, who have to 
 modify their light by all manner of screens and 
 curtains in order to obtain well-modelled portraits. 
 
 Of more importance for the representation of 
 depth than the elements hitherto enumerated, and 
 which are more or less of local and accidental signific- 
 ance, is what is called aerial perspective. By this we 
 understand the optical action of the light, which the 
 illuminated masses of air, between the observer and 
 distant objects, give. This arises from a fine opacity 
 
88 ON THE RELATION OF OPTICS TO PAINTING. 
 
 in the atmosphere, which never entirely disappears. 
 If, in a transparent medium, there are fine transparent 
 particles of varying density and varying refrangibility, 
 in so far as they are struck by it, they deflect the 
 light passing through such a medium, partly by reflec- 
 tion and partly by refraction ; to use an optical expres- 
 sion, they scatter it in all directions. If the opaque 
 particles are sparsely distributed, so that a great part 
 of the light can pass through them without being 
 deflected, distant objects are seen in sharp, well-defined 
 outlines through such a medium, while at the same 
 time a portion of the light which is deflected is dis- 
 tributed in the transparent medium as an opaque halo. 
 Water rendered turbid by a few drops of milk shows 
 this dispersion of the light and cloudiness very distinctly. 
 The light in this case is deflected by the microscopic 
 globules of butter which are suspended in the milk. 
 
 In the ordinary air of our rooms, this turbidity is 
 very apparent when the room is closed, and a ray of 
 sunlight is admitted through a narrow aperture. We 
 see then some of these solar particles, large enough to 
 be distinguished by the naked eye, while others form 
 a fine homogeneous turbidity. But even the latter 
 must consist mainly of suspended particles of organic 
 substances, for, according to an observation of Tyndall, 
 they can be burnt. If the flame of a spirit lamp is 
 placed directly below the path of these rays, the air 
 
ON THE RELATION OF OPTICS JO PAINTING. 89 
 
 rising from the flame stands out quite dark in the 
 surrounding bright turbidity ; that is to say, the air 
 rising from the flame has been quite freed from dust. 
 In the open air, besides dust and occasional smoke, we 
 must often also take into account the turbidity arising 
 from incipient aqueous deposits, where the tempera- 
 ture of moist air sinks so far that the water retained 
 in it can no longer exist as invisible vapour. Part of 
 the water settles then in the form of fine drops, as a 
 kind of the very finest aqueous dust, and forms a finer 
 or denser fog ; that is to say, cloud. The turbidity 
 which forms in hot sunshine and dry air may arise, 
 partly from dust which the ascending currents of 
 warm air whirl about; and partly from the irregular 
 mixture of cold and warm layers of air of different 
 density, as is seen in the tremulous motion of the 
 lower layers of air over surfaces irradiated by the sun. 
 But science can as yet give no explanation of the 
 turbidity in the higher regions of the atmosphere 
 which produces the blue of the sky ; we do not know 
 whether it arises from suspended particles of foreign 
 substances, or whether the molecules of air themselves 
 may not act as turbid particles in the luminous ether. 
 
 The colour of the light reflected by the opaque 
 particles mainly depends on their magnitude. When 
 a block of wood floats on water, and by a succession of 
 falling drops we produce small wave-rings near it. 
 
90 ON THE RELATION OF OPTICS TO PAINTING. 
 
 these are repelled by the floating wood as if it were a 
 solid wall. But in the long waves of the sea, a block 
 of wood would be rocked about without the waves 
 being thereby materially disturbed in their progress. 
 Now light is well known to be an undulatory motion 
 of the ether which fills all space. The red and yellow 
 rays have the longest waves, the blue and violet the 
 shortest. Very fine particles, therefore, which disturb 
 the uniformity of the ether, will accordingly reflect 
 the latter rays more markedly than the red and yellow 
 rays. The light of turbid media is bluer, the finer 
 are the opaque particles ; while the larger particles of 
 uniform light reflect all colours, and therefore give a 
 whitish turbidity. Of this kind is the celestial blue, 
 that is, the colour of the turbid atmosphere as seen 
 against dark cosmical space. The purer and the more 
 transparent the air, the bluer is the sky. In like man- 
 ner it is bluer and darker when we ascend high moun- 
 tains, partly because the air at great heights is freer 
 from turbidity, and partly because there is less air above 
 us. But the same blue, which is seen against the dark 
 celestial space, also occurs against dark terrestrial 
 objects ; for instance, when a thick layer of illuminated 
 air is between us and masses of deeply shaded or 
 wooded hills. The same aerial light makes the sky 
 blue, as well as the mountains ; excepting that in the 
 former case it is pure, while in the latter it is mixed 
 
ON THE RELATION OF OPTICS TO PAINTING. 91 
 
 with the light from objects behind; and moreover 
 it belongs to the coarser turbidity of the lower regions 
 of the atmosphere, so that it is whiter. In hot coun- 
 tries, and with dry air, the aerial turbidity is also finer 
 in 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 snow-mountains, appear coloured. 
 
92 ON THE RELATION OF OPTICS TO PAINTING. 
 
 These colourations are moreover not peculiar to 
 the air, but occur in all cases in which a transparent 
 substance is made turbid by the admixture of another 
 transparent substance. We see it, as we have ob- 
 served, in diluted milk, and in water to which a few 
 drops of eau de Cologne have been added, whereby the 
 ethereal oils and resins dissolved by the latter, sepa- 
 rate out and produce the turbidity. Excessively fine 
 blue clouds, bluer even than the air, may be produced, 
 as Tyndall has observed, when the sun's light is 
 allowed to exert its decomposing action on the vapours 
 of certain carbon compounds. Groethe called attention 
 to the universality of this phenomenon, and endea- 
 voured to base upon it his theory of colour. 
 
 By aerial perspective we understand the artistic 
 representation of aerial turbidity ; for the greater or 
 less predominance of the aerial colour above the colour 
 of the objects, shows their varying distance very 
 definitely ; and landscapes more especially acquire the 
 appearance of depth. According to the weather, the 
 turbidity of the air may be greater or less, more white 
 or more blue. Very clear air, as sometimes met with 
 after continued rain, makes the distant mountains 
 appear small and near ; whereas, when the air contains 
 more vapour, they appear large and distant. 
 
 This latter is decidedly better for the landscape 
 painter, and the high transparent landscapes of moun- 
 
ON THE RELATION OF OPTICS TO PAINTING. 93 
 
 tainous regions, which so often lead the Alpine climber 
 to under-estimate the distance and the magnitude of 
 the mountain-tops before him, are also difficult to turn 
 to account in a picturesque manner. Views from the 
 valleys, and from seas and plains in which the aerial 
 light is faintly but markedly developed, are far better ; 
 not only do they allow the various distances and mag- 
 nitudes of what is seen to stand out, but they are on 
 the other hand favourable to the artistic unity of 
 colouration. 
 
 Although aerial colour is most distinct in the 
 greater depths of landscape, it is not entirely wanting 
 in front of the near objects of a room. What is seen 
 to be isolated and well defined, 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. 
 
 LIBRA n i 
 
 UNI VKKS1TY Ob 
 
 i 1 T I > 4 \ 1 ) \ T I \ 
 
94 ON THE KELATION OF OPTICS TO PAINTING. 
 
 II. SHADE. 
 
 The circumstances which we have hitherto dis- 
 cussed indicate a profound difference, and one which is 
 exceedingly important for the perception of solid form, 
 between the visual image which our eyes give, when we 
 stand before objects, and that which the picture gives. 
 The choice of the objects to be represented in pictures 
 is thereby at once -much restricted. Artists are well 
 aware that there is much which cannot be represented 
 by the means at their disposal. Part of their artistic 
 skill consists in the fact that by a suitable grouping, 
 position, and turn of the objects, by a suitable choice 
 of the point of view, and by the mode of lighting, 
 they learn to overcome the unfavourable conditions 
 which are imposed on them in this respect. 
 
 It might at first sight appear that of the requisite 
 truth to nature of a picture, so much would remain 
 that, seen from the proper point of view, it would at 
 least produce the same distribution of light, colour, 
 and shadow in its field of view, and would produce in 
 the interior of the eye exactly the same image on the 
 retina as the object represented would do if we had it 
 actually before us, and looked at it from a definite, 
 
ON THE KELATION OF OPTICS TO PAINTING. 95 
 
 fixed point of view. It might seem to be an object 
 of pictorial skill to aim at producing, under the given 
 limitations, the same effect as is produced by the 
 object itself. 
 
 If we proceed to examine whether, and how far, 
 painting can satisfy such a condition, we come upon 
 difficulties before which we should perhaps shrink, if 
 we did not know that they had been already over- 
 come. 
 
 Let us begin with the simplest case ; with the quan- 
 titative relations between luminous intensities. If the 
 artist is to imitate exactly the impression which the 
 object produces on our eye, he ought to be able to 
 dispose of brightness and darkness equal to that which 
 nature offers. But of this there can be no idea. Let 
 me give a case in point. Let there be, in a pic- 
 ture-gallery, a desert-scene, in which a procession of 
 Bedouins, shrouded in white, and of dark negroes, 
 marches under the burning sunshine; close to it a 
 bluish moonlight scene, where the moon is reflected in 
 the water, and groups of trees, and human forms, are 
 seen to be faintly indicated in the darkness. You 
 know from experience that both pictures, if they 
 are well done, can produce with surprising vividness 
 the representation of their objects; and yet, in both 
 pictures, the brightest parts are produced with the 
 same white-lead, which is but slightly altered by ad- 
 
96 ON THE KELATION OF OPTICS TO PAINTING. 
 
 mixtures; while the darkest parts are produced with 
 the same black. Both, being hung on the same wall, 
 share the same light, and the brightest as well as the 
 darkest parts of the two scarcely differ as concerns 
 the degree of their brightness. 
 
 How is it, however, with the actual degrees of 
 brightness represented ? The relation between the 
 brightness of the sun's light, and that of the moon, 
 was measured by Wollaston, who compared their in- 
 tensities with that of the light of candles of the same 
 material. He thus found that the luminosity of the 
 sun is 800,000 times that of the brightest light of a 
 full moon. 
 
 An opaque body, which is lighted from any source 
 whatever, can, even in the most favourable case, only 
 emit as much light as falls upon it. Yet, from Lam- 
 bert's observations, even the whitest bodies only reflect 
 about two fifths of the incident light. The sun's rays, 
 which proceed parallel from the sun, whose diameter 
 is 85,000 miles, when they reach us, are distributed 
 uniformly over a sphere 195 millions of miles in dia- 
 meter. Its density and illuminating power is here 
 only the one forty-thousandth of that with which it 
 left the sun's surface; and Lambert's number leads to 
 the conclusion that even the brightest white surface 
 on which the sun's rays fall vertically, has only the 
 one hundred-thousandth part of the brightness of the 
 
ON THE RELATION OF OPTICS TO PAINTING. 97 
 
 sun's disk. The moon however is a gray body, whose 
 mean brightness is only about one fifth of that of the 
 purest white. 
 
 And when the moon- irradiates a body of the purest 
 white on the earth, its brightness is only the hundred- 
 thousandth part of the brightness of the moon itself ; 
 hence the sun's disk is 80,000 million times brighter 
 than a white which is irradiated by the full moon. 
 
 Now pictures which hang in a room are not lighted 
 by the direct light of the sun, but by that which is re- 
 flected from the sky and clouds. I do not know of any 
 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 l-20th 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. I found in fact, by an experiment, that lamp- 
 
 II. H 
 
98 ON THE RELATION OF OPTICS TO PAINTING. 
 
 black, lighted by the sun, is not less than half as- 
 bright, as shaded white in the brighter part of a 
 room. 
 
 On the picture of the moon, the same white which 
 has been used for depicting the Bedouins' garments 
 must be used for representing the moon's disk, and its 
 reflection in the water ; although the real moon has 
 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 y-J-^ that of white paper. The brightest 
 colours of a painter are only about one hundred times 
 as bright as his darkest shades. 
 
 The statements I have made may perhaps appear 
 exaggerated. But they depend upon measurements, 
 and you can control them by well-known observations. 
 According to Wollaston, the light of the full moon is 
 
ON THE RELATION OF OPTICS TO PAINTING. 99 
 
 equal to that of a candle burning at a distance of 12 
 feet. You know that we cannot read by the light of the 
 full moon, though we can read at a distance of three or 
 four feet from a candle. Now assume that you suddenly 
 passed from a room in daylight to a vault perfectly 
 dark, with the exception of the light of a single candle. 
 You would at first think you were in absolute darkness, 
 and at most you would only recognise the candle itself. 
 In any case, you would not recognise the slightest trace 
 of any objects at a distance of 12 feet from the candle. 
 These however are the objects whose illumination is 
 the same as that which the moonlight gives. You 
 would only become accustomed to the darkness after 
 some time, and you would then find your way about 
 without difficulty. 
 
 If, now, you return to the daylight, which before 
 was perfectly comfortable, it will appear so dazzling that 
 you will perhaps have to close the eyes, and only be 
 able to gaze round with a painful glare. You see 
 thus that we are concerned here not with minute, but 
 with colossal, differences. How now is it possible that, 
 under such circumstances, we can imagine there is any 
 similarity between the picture and reality ? 
 
 Our discussion of what we did not see at first, but 
 could afterwards see in the vault, points to the most 
 important element in the solution ; it is the varying 
 extent to which our senses are deadened by light ; a 
 
 H 2 
 
100 ON THE RELATION OF OPTICS TO PAINTING. 
 
 process to which we can attach the same name, fatigue, 
 as that for the corresponding one in the muscle. Any 
 activity of our nervous system diminishes its power for 
 the time being. The muscle is tired by work, the 
 brain is tired by thinking, and by mental operations ; 
 the eye is tired by light, and the more so the more 
 powerful the light. Fatigue makes it dull and in- 
 sensitive to new impressions, so that it appreciates 
 strong ones only moderately, and weak ones not at all. 
 But now you see how different is the aim of the 
 artist when these circumstances are taken into account. 
 The eye of the traveller in the desert, who is looking 
 at the caravan, has been dulled to the last degree by the 
 dazzling sunshine ; while that of the wanderer by moon- 
 light has been raised to the extreme of sensitiveness. 
 The condition of one who is looking at a picture 
 differs from both the above cases by possessing a cer- 
 tain mean degree of sensitiveness. Accordingly, the 
 painter must endeavour to produce by his colours, on 
 the moderately sensitive eye of the spectator, the same 
 impression as that which the desert, on the one hand, 
 produces on the deadened, and the moonlight, on the 
 other hand, creates on the untired eye of its observer. 
 Hence, along with the actual luminous phenomena of 
 the outer world, the different physiological conditions 
 of the eye play a most important part in the work of 
 the artist. What he has to give is not a mere tran- 
 
ON THE RELATION OF OPTICS TO PAINTING. 101 
 
 script of the object, but a translation of his impression 
 into another scale of sensitiveness, which belongs to a 
 different degree of impressibility of the observing eye, 
 in which the organ speaks a very different dialect in 
 responding to the impressions of the outer world. 
 
 In order to understand to what conclusions this 
 leads, I must first of all explain the law which Fechner 
 discovered for the scale of sensitiveness of the eye, 
 which is a particular case of the more general psycho- 
 physical laiu of the relations of the various sensuous 
 impressions to the irritations which produce them. This 
 law may be expressed as follows : Within very wide 
 limits of brightness, differences in the strength of light 
 are equally distinct or appear equal in sensation, if 
 they form an equal fraction of the total quantity of 
 light compared. Thus, for instance, differences in in- 
 tensity of one hundredth of the total amount can be 
 recognised without great trouble with very different 
 strengths of light, without exhibiting material dif- 
 ferences in the certainty and facility of the estimate, 
 whether the brightest daylight or the light of a good 
 candle be used. 
 
 The easiest method of producing accurately mea- 
 surable differences in the brightness of two white 
 surfaces, depends on the use of rapidly rotating disks, 
 If a disk, like the adjacent one in Fig. 3, is made to 
 rotate very rapidly (that is, 20 to 30 times in a second), 
 
102 ON THE EELATION OF OPTICS TO PAINTING. 
 
 it appears to the eye to be covered Avith three grey 
 rings as in Fig. 4. The reader must, however, figure 
 to himself the grey of these rings, as it appears on 
 
 FIG. 3. FIG. 4. 
 
 the rotating disk of Fig. 3, as a scarcely perceptible 
 shade of the ground. When the rotation is rapid 
 each ring of the disk appears illuminated, as if all the 
 light which fell upon it had been uniformly distributed 
 'Over its entire surface. Those rings, in which are the 
 black bands, have somewhat less light than the quite 
 white ones, and if the breadth of the marks is com- 
 pared with the length of half the circumference of the 
 corresponding ring, we get the fraction by which the 
 intensity of the light in the white ground of the disk is 
 diminished in the ring in question. If the bands are all 
 equally broad, as in Fig. 3, the inner rings appear darker 
 than the outer ones, for in this latter case the same 
 loss of light is distributed over a larger area than in 
 the former. In this way extremely delicate shades of 
 
ON THE RELATION OF OPTICS TO PAINTING. 103 
 
 "brightness may be obtained, and by this method, when 
 the strength of the illumination varies, the brightness 
 always diminishes by the same proportion of its total 
 value. Now it is found, in accordance with Fechner's 
 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 
 
104 ON THE KELATION OF OPTICS TO PAINTING. 
 
 difference between daylight, on the one hand, and day- 
 light plus candle-light on the other, can be no longer 
 recognised. 
 
 This law is of great importance in discriminating 
 between various degrees of brightness of natural objects, 
 A white body appears white because it reflects a large 
 fraction, and a grey body appears grey because it re- 
 flects a small fraction, of incident light. For different 
 intensities of illumination, the difference of brightness 
 between the two will always correspond to the same frac- 
 tion of their total brightness, and hence will be equally 
 perceptible to our eyes, provided we do not approach too 
 near to the upper or the lower limit of the brightness, 
 for which Fechner's law no longer holds. Hence, on 
 the whole, the painter can produce what appears an equal 
 difference for the spectator of his picture, notwithstand- 
 ing the varying strength of light in the gallery, provided 
 he gives to his colours the same ratio of brightness as 
 that which actually exists. 
 
 For, in fact, in looking at natural objects, the abso- 
 lute brightness in which they appear to the eye varies 
 within very wide limits, according to the intensity of 
 the light, and the sensitiveness of the eye. That which 
 is constant is only the ratio of the brightness in which 
 surfaces of various depth of colour appear to us when 
 lighted to the same amount. But this ratio of bright- 
 ness is for us the perception, from w T hich we form our 
 
ON THE RELATION OF OPTICS TO PAINTING. 105 
 
 judgment as to 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 y 
 in like manner also, nations in which painting has- 
 remained in a childish stage. Perfect artistic painting 
 is only reached when we have succeeded in imitating 
 the action of light upon the eye, and not merely the 
 pigments; and only when we look at the object of 
 pictorial representation from this point of view, will it 
 be possible to understand the variations from nature 
 which artists have to make in the choice of their scale 
 of colour and of shade. 
 
 These are, in the first case, due to the circumstance 
 that Fechner's law only holds for mean degrees of 
 
106 ON THE EELATION OF OPTICS TO PAINTING. 
 
 brightness ; while, for a brightness which is too high 
 <or too low, appreciable divergences are met with. 
 
 At both extremes of luminous intensity the eye is 
 less sensitive for differences in light than is required by 
 that law. With a very strong light it is dazzled ; that 
 is, its internal activity cannot keep pace with the ex- 
 ternal excitations ; the nerves are too soon tired. Very 
 bright objects appear almost always to be equally 
 bright, even when there are, in fact, material differ- 
 ences in their luminous intensity. The light at the 
 edge of the sun is only about half as bright as that at 
 the centre, yet none of you will have noticed that, if 
 you have not looked through coloured glasses, which 
 reduce the brightness to a convenient extent. With 
 a weak light the eye is also less sensitive, but from the 
 opposite reason. If a body is so feebly illuminated 
 that we scarcely perceive it, we shall not be able to 
 perceive that its brightness is lessened by a shadow 
 by the one hundredth or even by a tenth. 
 
 It follows from this, that, with moderate illumina- 
 tion, darker objects become more like the darkest 
 objects, while w^ith greater illumination brighter ob- 
 jects become more like the brightest than should be 
 the case in accordance with Fechner's law, which 
 holds for mean degrees of illumination. From this 
 results, what, for painting, is an extremely characteristic 
 
ON THE RELATION OF OPTICS TO PAINTING. 107 
 
 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 
 ^ye 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 too bright or too feeble lights. If 
 they could employ the colour of the dazzling bright- 
 ness of full sunshine, or of the actual dimness of 
 moonlight, they would not need to represent the 
 gradation of light in their picture other than it is in 
 nature; the picture would then make the same im- 
 pression on the eye as is produced by equal degrees of 
 brightness of actual objects. The alteration in the 
 scale of shade which has been described is necessary 
 because the colours of the picture are seen in the 
 mean brightness of a moderately lighted room, for 
 
108 ON THE RELATION OF OPTICS TO PAINTING. 
 
 which Fechner's 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 too 1 
 
ON THE RELATION OF OPTICS TO PAINTING. 109 
 
 low lights, are not without some influence in the region 
 of the middle lights. We have to observe more closely 
 in order to perceive this influence. It is found, in fact, 
 that when the very finest differences of shade are pro- 
 duced on a rotating disk, they are only visible by a 
 light which about corresponds to the illumination of a 
 white paper on a bright day, which is lighted by the 
 light of the sky, but is not directly struck by the 
 sun. With such a light, shades of y^ or yi~o f 
 the total intensity can be recognised. The light in 
 which pictures are looked at is, on the contrary, much 
 feebler ; and if we are to retain the same distinctness 
 of the finest shadows and of the modelling of the 
 contours which it produces, the gradations of shade 
 in the picture must be somewhat stronger than cor- 
 responds to the exact luminous intensities. The 
 darkest objects of the picture thereby become un- 
 naturally dark, which is however not detrimental to 
 the object of the artist if the attention of the observer 
 is to be directed to the brighter parts. The great 
 artistic effectiveness of this manner shows us that the 
 chief emphasis is to be laid on imitating difference of 
 brightness and not on absolute brightness ; and that the 
 greatest differences in this latter respect can be borne 
 without perceptible incongruity, if only their grada- 
 tions are imitated with expression. 
 
110 ON THE RELATION OF OPTICS TO PAINTING. 
 
 III. COLOUR. 
 
 With these divergences in brightness are connected 
 certain divergences in colour, which, physiologically, 
 are caused by the fact that the scale of sensitiveness 
 is different for different colours. The strength of the 
 sensation produced by light of a particular colour, and 
 for a given intensity of light, depends altogether on 
 the special reaction of that complex of nerves which 
 are set in operation by the action of the light in 
 question. Now all our sensations of colour are ad- 
 mixtures of three simple sensations ; namely, of red, 
 green, and violet, 1 which, by a not improbable suppo- 
 sition of Thomas Young, can be apprehended quite 
 independently of each other by three different systems 
 of nerve-fibres. To this independence of the different 
 sensations of colour corresponds their independence in 
 the gradation of intensity. Kecent 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 -jW to -^-J-g of the intensity 
 can be observed in the blue, and with an untired eye 
 
 1 Helmholtz's Popular Scientific Leuturvs, pp. 232-52. 
 
 2 Dobrowolsky in GTdi'fe's Arcldr fiir Ojththstlntologie, vol. xviii, 
 part i. pp. 24-92. 
 
ON THE RELATION OF OPTICS TO PAINTING. 11| 
 
 of -jig- in the red ; or when the colour is dimmed 
 by being looked at for a long time, a difference of 
 
 3V to TV 
 
 Red therefore acts as a colour towards whose shades 
 the eye is 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 y 
 and even in a more striking manner, in the red and 
 violet spectral colours, and, when their intensity is- 
 increased only moderately, by the same fraction for 
 both. 
 
 Now the impression of white is made up of the 
 impressions which the individual spectral colours make 
 on our eye. If we increase the brightness of white, 
 the strength of the sensation for the red and yellow 
 rays will relatively be more increased than that for 
 the blue and violet. In bright white, therefore, the 
 former will produce a relatively stronger impression 
 than the latter; in dull white the blue and bluish 
 colours will have this effect. Very bright white appears 
 therefore yellowish, and dull white appears bluish. In 
 
112 ON THE RELATION OF OPTICS TO PAINTING. 
 
 our ordinary way of looking at the objects about us, 
 we are not so readily conscious of this ; for the direct 
 comparison of colours of very different shade is diffi- 
 cult, and we are accustomed to see in this alteration in 
 the white the result of different illumination of one and 
 the same white object, so that in judging pigment- 
 colours we have learnt to eliminate the influence of 
 brightness. 
 
 If however to the painter is put the problem of imi- 
 tating, with faint colours, white irradiated by the sun, 
 he can attain a high degree of resemblance ; for by an 
 admixture of yellow in his white he makes this colour 
 preponderate just as it would preponderate in actual 
 bright light, owing to the impression on the nerves. 
 It is the same impression as that produced if we look 
 at a clouded landscape through a yellow glass, and 
 thereby give it the appearance of a sunny light. The 
 artist will, on the contrary, give a bluish tint to moon- 
 light, that is, a faint white ; for the colours on the 
 picture must, as we have seen, be far brighter than 
 the colour to be represented. In moonshine scarcely 
 any other colour can be recognised than blue ; the 
 blue starry sky or blue colours may still appear 
 distinctly coloured, while yellow and red can only be 
 seen as obscurations of the general bluish white or 
 
 I will again remind you that these changes of 
 
ON THE RELATION OF OPTICS TO PAINTING. 113 
 
 colour would 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, 
 H. I 
 
114 ON THE RELATION OF OPTICS TO PAINTING. 
 
 however, enter upon them, for they produce no diver- 
 gence between the picture and reality. 
 
 The second class of phenomena of contrast, and one 
 which, for us, is more important, is met with in 
 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, denned picture it 
 can also be locally developed in this part only. 
 
OX THE RELATION OF OPTICS TO PAINTING. 115 
 
 You must all have observed the dark spots which 
 move about in the field of vision, when we have, been 
 looking for only a short time towards the setting sun, 
 and which physiologists call negative after-images of 
 the sun. They are due to the fact that only those parts 
 of the retina which are actually struck by the image of 
 the sun in the eye, have become insensitive to a new 
 impression of light. If, with an eye which is thus 
 locally tired, we look towards a uniformly bright sur- 
 face, such as the sky, the tired parts of the retina are 
 more feebly and more darkly affected than the other 
 portions, so that the observer thinks he sees dark spots 
 in the sky, which move about with his sight. We 
 have then in juxtaposition, in the bright parts of 
 the sky, the impression which these make upon the 
 untired parts of the retina, and in the dark spots 
 their action on the tired portions. Objects, bright 
 like the sun, produce negative after-images in the 
 most striking manner; but with a little attention they 
 may be seen even after much more moderate impres- 
 sions of light. A longer time is required in order to de- 
 velop such an impression, so that it may be distinctly 
 recognised, and a definite point of the bright object 
 must be fixed, without moving the eye, so that its image 
 may be distinctly formed on the retina, and only a 
 limited portion of the retina be excited and tired, 
 just as in producing sharp photographic portraits, the 
 
 i 2 
 
116 ON THE RELATION OF OPTICS TO PAINTING. 
 
 object must be stationary during the time of exposure 
 in order that its image may not be displaced on the 
 sensitive plate. The after-image in the eye is, as it 
 were, a photograph on the retina, which becomes 
 visible owing to the altered sensitiveness towards 
 fresh light, but only remains stationary for a short 
 time ; it is longer, the more powerful and durable was 
 the action of light. 
 
 If the object viewed was coloured, for instance 
 red paper, the after-image is of the complementary 
 colour on a grey ground ; in this case of a bluish green. 1 
 Rose-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 minute or two, the cross being steadily viewed, and it is then 
 drawn suddenly away, without relaxing the view. In place of the 
 sheet removed the after-image appears then on the dark ground. 
 
 2 See Helmholtz's Popular Lectures, first series, p. 250. 
 
ON THE RELATION OF OPTICS TO PAINTING. 117 
 
 If this same part of the retina is afterwards illuminated 
 with white light, the sensation of green is enfeebled, 
 while that of red and violet is vivid and predominant ; 
 their sum gives the sensation of purple, which mixed 
 with the unchanged white ground forms rose-red. 
 
 In the ordinary way of looking at light and coloured 
 objects, we are not accustomed to fix continuously one 
 and the same point ; for following with the gaze the 
 play of our attentiveness, we are always turning it to 
 new parts of the object as they happen to interest us. 
 This way of looking, in which the eye is continually 
 moving, and therefore the retinal image is also shift- 
 ing about on the retina, has moreover the advantage 
 of avoiding disturbances of sight, which powerful and 
 continuous after-images would bring with them. Yet 
 here also, after-images are not wanting ; only they are 
 shadowy in their contours, and of very short duration. 
 
 If a red surface be laid upon a grey ground, and if 
 we look from the red over the edge towards the grey, 
 the edges of the grey will seem as if struck by such an 
 after-image of red, and will seem to be of a faint 
 bluish green. But as the after-image rapidly disappears, 
 it is mostly only those parts of the grey, which are nearest 
 the red, which show the change in a marked degree. 
 
 This also is a phenomenon which is produced more 
 strongly by bright light and brilliant saturated colours 
 than by fainter light and duller colours. The artist, 
 
118 ON THE RELATION OF OPTICS TO PAINTING. 
 
 however, works for the most part with the latter. He 
 produces most of his tints by mixture ; each mixed 
 pigment is, however, greyer and duller than the pure 
 colour of which it is mixed, and even the few pig- 
 ments of a highly saturated shade, which oil-painting 
 can employ, are comparatively dark. The pigments 
 employed in water-colours and coloured chalks are 
 again comparatively white. Hence such bright con- 
 trasts, as are observed in strongly coloured and strongly 
 lighted objects in nature, cannot be expected from 
 their representation in the picture. If, therefore, 
 with the pigments at his command, the artist wishes 
 to reproduce the impression which objects give, as 
 strikingly as possible, he must paint the contrasts 
 which they produce. If the colours on the picture 
 are as brilliant and luminous as in the actual objects, 
 the contrasts in the former case would produce them- 
 selves as spontaneously as in the latter. Here, also, 
 subjective phenomena of the eye must be objectively 
 introduced into the picture, because the scale of colour 
 and of brightness is different upon the latter. 
 
 With a little attention you will see that painters 
 and draughtsmen generally make a plain uniformly 
 lighted surface brighter, where it is close to a dark 
 object, and darker, where it is near a light object. 
 You will find that uniform grey surfaces are given 
 a yellowish tint at the edge where there is a back- 
 
ON THE RELATION OF OPTICS TO PAINTING. 119 
 
 ground of blue, and a rose-red tint where they im- 
 pinge 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 brignt 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 
 
120 ON THE RELATION OF OPTICS TO PAINTING. 
 
 part of the field of view are seen more distinctly. If 
 the flame is partly screened by a ruler, this appears 
 jagged where the flame projects beyond it. The lu- 
 minosity in the neighbourhood of the flame is so in- 
 tense, that its brightness can scarcely be distinguished 
 from that of the flame itself; as is the case with all 
 bright objects, the flame appears magnified, and as if 
 spreading over towards the adjacent dark objects. 
 
 The cause of this phenomenon is quite similar to 
 that of aerial perspective. It is due to a diffusion of 
 light which arises from the passage of light through 
 dull media, excepting that for the phenomena of aerial 
 perspective the turbidity is to be sought in the air in 
 front of the eye, while for true phenomena of irradiation 
 it is to be sought in the transparent media of the eye. 
 When 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 are 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 
 1 I disregard here the view that irradiation in the eye depends on 
 
ON THE KELATION OF OPTICS TO PAINTING. 121 
 
 The phenomena of irradiation also occur with 
 moderate degrees of brightness. A dark aperture 
 in a sheet of paper illuminated by the sun, or a small 
 dark object on a coloured glass plate which is held 
 against the clear sky, appear as if the colour of the 
 adjacent surface were diffused over them. 
 
 Hence the phenomena of irradiation are very similar 
 to those which produce the opacity of the air. The 
 only essential difference lies in this, that the opacity 
 by luminous air is stronger before distant objects which 
 have a greater mass of air in front of them than before 
 near ones ; while irradiation in the eye sheds its halo 
 uniformly over near and over distant objects. 
 
 Irradiation also belongs to the subjective pheno- 
 mena of the eye which the artist represents objectively, 
 because painted lights and painted sunlight are not 
 bright enough to produce a distinct irradiation in the 
 eye of the observer. 
 
 The representation which the painter has to give 
 of the lights and colours of his object I have described 
 as a translation, and I have urged that, as a general 
 rule, it cannot give a copy true in all its details. The 
 altered scale of brightness which the artist must 
 apply in many cases is opposed to this. It is not the 
 colours of the objects, but the impression which they 
 
 a diffusion of the excitation in the substance of the nerves, as this 
 appears to me too hypothetical. Moreover, we are here concerned 
 with the phenomena and not with their cause. 
 
122 ON THE RELATION OF OPTICS TO PAINTING. 
 
 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 Rembrandt, he may exaggerate them 
 
OX THE RELATION OF OPTICS TO PAINTING. 123 
 
 in order to obtain strong relief ; or he may diminish 
 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 
 ' style' or 'treatment,' and indeed in their purely pic- 
 torial elements. 
 
 I B R A R V 
 
 UNIVKKSITY OF 
 
 ALIFORNIA: 
 
124 ON THE KELATION OF OPTICS TO PAINTING. 
 
 IV. HARMONY OF COLOUR. 
 
 We here naturally raise the question : If, owing to the 
 small quantity of light and saturation of his colours, 
 the artist seeks, in all kinds of indirect ways, by imi- 
 tating subjective impressions to attain resemblance to 
 nature, as close as possible, but still imperfect, would 
 it not be more convenient to seek for means of obvi- 
 ating these evils ? Such there are indeed. Frescoes 
 are sometimes viewed in direct sunlight ; transparen- 
 cies and paintings on glass can 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 
 artistic effects could be attained with colours 
 lighted by the sun, we should undoubtedly have pic- 
 tures which took advantage of this. Fresco painting 
 
OX THE RELATION OF OPTICS TO PAINTING. 125 
 
 would 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 
 Munich, to learn in what this advantage consists. 
 Their brightness is so great that we cannot look at 
 them steadily for any length of time. And what in 
 this case is so painful and so tiring to the eye, would 
 also operate in a smaller degree if, in a picture, bril- 
 liant colours were used, even locally and to a moderate 
 extent, which were intended to represent bright sun- 
 light, and a mass of light shed over the picture. 
 It is much easier to produce an accurate imitation 
 of the feeble light of moonshine with artificial light 
 in dioramas and theatre decorations. 
 
 We may therefore designate truth to Nature of a 
 beautiful picture as an ennobled fidelity to Nature. 
 Such a picture reproduces all that is essential in the 
 impression, and attains full vividness of conception, 
 but without injury or tiring the eye by the nude lights 
 of reality. The differences between Art and Nature 
 are chiefly confined, as we have already seen, to those 
 
126 ON THE RELATION OF OPTICS TO PAINTING. 
 
 matters which, we can in reality only estimate in an un- 
 certain manner, such as the absolute intensities of light. 
 
 That which is pleasant to the senses, the beneficial 
 but not exhausting fatigue of our nerves, the feeling 
 of comfort, corresponds in this case, as in others, to 
 those conditions which are most favourable for per- 
 ceiving the outer world, and which admit of the finest 
 discrimination and observation. 
 
 It has been mentioned above that the discrimina- 
 tion of the finest shadows, and of the modelling which 
 they express, is the most delicate under a certain 
 mean brightness. I should like to direct your atten- 
 tion to another point which has great importance in 
 painting: I refer to our natural delight in colours, 
 which has undoubtedly a great influence upon our 
 pleasure in the works of the painter. In its simplest 
 expression, as pleasure in gaudy flowers, feathers, 
 stones, in fireworks, and Bengal lights, this inclination 
 has but little to do with man's sense of art ; it only ap- 
 pears as the natural pleasure of the perceptive organism 
 in the varying and multifarious excitation of its various 
 nerves, which is necessary for its healthy continuance 
 and productivity. But the thorough fitness in the con- 
 struction of living organisms, whatever their origin, 
 excludes the possibility that in the majority of healthy 
 individuals an instinct should be developed or main- 
 tain itself which did not serve some definite purpose.' 
 
ON THE RELATION OF OPTICS TO PAINTING. 127 
 
 We have not far to seek for the delight in light 
 and in colours, and for the dread of darkness; this 
 coincides with the endeavour to see and to recognise 
 surrounding objects. Darkness owes the greater part 
 of the terror which it inspires to the fright of what 
 is unknown and cannot be recognised. A coloured 
 picture gives a far more accurate, richer, and easier 
 conception than a similarly executed drawing, which 
 only retains the contrasts of light and shade. A 
 picture retains the latter, but has in addition the 
 material for discrimination which colours afford; by 
 which surfaces which appear equally bright in the 
 drawing, owing to their different colour, are now 
 assigned to various objects, or again as alike in colour 
 are seen to be parts of the same, or of similar objects. 
 In utilising the relations thus naturally given, the 
 artist, by means of prominent colours, can direct and 
 enchain the attention of the observer upon the chief 
 objects of the picture; and by the variety of the 
 garments he can discriminate the figures from each 
 other, but complete each individual one in itself. 
 Even the natural pleasure in pure, strongly saturated 
 colours, finds its justification in this direction. The 
 case is analogous to that in music, with the full, pure, 
 well-sounding tones of a beautiful voice. Such a one 
 is more expressive ; that is, even the smallest change 
 of its pitch, or its quality any slight interruption, 
 
128 ON THE EELATION OF OPTICS TO PAINTING. 
 
 any tremulousness, any rising or falling in it is at 
 once more distinctly recognised by the hearer than 
 could be the case with a less regular sound ; and it 
 seems also that the powerful excitation which it pro- 
 duces in the ear of the listener, arouses trains of ideas 
 and passions more strongly than does a feebler excita- 
 tion of the same kind. A pure, fundamental colour 
 bears to small admixtures the same relation as a dark 
 ground on which the slightest shade of light is visible. 
 Any of the ladies present will have known how sensi- 
 tive clothes of uniform saturated shades are to dirt, 
 in comparison with grey or greyish-brown materials. 
 This also corresponds to the conclusions from Young's 
 theory of colours. According to this theory, the per- 
 ception of each of the three fundamental colours 
 arises from the excitation of only one kind of sensitive 
 fibres, while the two others are at rest ; or at any rate 
 are but feebly excited. A brilliant, pure colour pro- 
 duces a powerful stimulus, and yet, at the same time, 
 a great degree of sensitiveness to the admixture of 
 other colours, in those systems of nerve-fibres which 
 are at rest. The modelling of a coloured surface 
 mainly depends upon the reflection of light of other 
 colours which falls upon them from without. It is 
 more particularly when the material glistens that the 
 reflections of the bright places are preferably of the 
 colour of the incident light. In the depth of the 
 
ON THE RELATION OF OPTICS TO PAINTING. 129 
 
 folds, on the contrary, the 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 they 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 
 
 II. K 
 
130 ON THE EELATION OF OPTICS TO PAINTING. 
 
 the use of the pure colours, otherwise they distract 
 the attention, and the picture becomes glaring. It 
 is necessary, on the other hand, to avoid a onesided 
 fatigue of the eye by too prominent a colour. This is 
 effected either by introducing the prominent colour 
 to a moderate extent upon a dull, slightly coloured 
 ground, or by the juxtaposition of variously saturated 
 colours, which produce a certain equilibrium of irrita- 
 tion in the eye, and, by the contrast in their after- 
 images, strengthen and increase each other. A green 
 surface on which the green after-image of a purple one 
 falls, appears to be a far purer green than without 
 such an after-image. By fatigue towards purple, that 
 is towards red and violet, any admixture of these two 
 colours in the green is enfeebled, while this itself pro- 
 duces its full effect. In this way the sensation of 
 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 are 
 detrimental to each other, and acquire a grey tint. 
 
 These relations of the colours to each other have 
 manifestly a great influence on the degree of pleasure 
 which different combinations of colours afford. Two 
 
ON THE RELATION OF OPTICS TO PAINTING. 131 
 
 colours may, without injury, be juxtaposed, which 
 indeed are so similar as to look like varieties of the 
 same colour, produced by varying degrees of light and 
 shade. Thus, upon scarlet the more shaded parts ap- 
 pear of a carmine, or on a straw-colour they appear 
 of a golden yellow. 
 
 If we pass beyond these limits, we arrive at un- 
 pleasant combinations, such as carmine and orange, or 
 orange and straw-yellow. The distance of the colours 
 must then be increased, so as to create pleasing com- 
 binations once more. The complementary colours are 
 those which are most distant from each other. When 
 these are combined, such, for instance, as straw-colour 
 and ultramarine, or verdigris and purple, they have 
 something insipid but crude ; perhaps because we are 
 prepared to expect the second colour to appear as an 
 after-image of the first, and it does not sufficiently 
 appear to be a new and independent element in the 
 compound. Hence, on the whole, combinations of 
 those pairs are most pleasing in which the second 
 colour of the complementary tint is near the first, 
 though with a distinct difference. Thus, scarlet and 
 greenish blue are complementary. The combination 
 produced when the greenish blue is allowed to glide 
 either into ultramarine, or yellowish green (sap green), 
 is still more pleasing. In the latter case, the com- 
 bination tends towards yellow, and in the former, 
 
 x 2 
 
132 ON THE RELATION OF OPTICS TO PAINTING. 
 
 towards rose-red. Still more satisfactory combinations 
 are those of three tints which bring about equilibrium 
 in the impression of colour, and, notwithstanding the 
 great body of colour, avoid a onesided fatigue of the 
 eye, without falling into the baldness of complemen- 
 tary tints. To this belongs the combination which 
 the Venetian masters used so much red, green, and 
 violet; as well as Paul Veronese's purple, greenish 
 blue, and yellow. The former triad corresponds ap- 
 proximately to the three fundamental colours, in so 
 far as these can be produced by pigments ; the latter 
 gives the mixtures of each pair of fundamental colours. 
 It is however to be observed, that it has not yet been 
 possible to establish rules for the harmony of colours 
 with the same precision and certainty as for the con- 
 sonance of tones. On the contrary, a consideration of 
 the facts shows that a number of accessory influences 
 come into play, 1 when once the coloured surface is 
 also to produce, either wholly or in part, a representa- 
 tion of natural objects or of solid forms, or even if it 
 only offers a resemblance with the representation of 
 a relief, of shaded and of non-shaded surfaces. It 
 is moreover often difficult to establish, as a matter o f 
 fact, what are the colours which produce the harmonic 
 impression. This is pre-eminently the case with 
 
 1 Conf. E. Briicke, Die Physioloc/ie der Farben fur die Zrvecltc 
 der Kungtgemer'be. Leipzig, 1866. W. v. Bezold, Die Farlenlelire 
 in Hinblick auf Kunst wid Kunstgenerbe. Braunschweig, 1874. 
 
ON THE RELATION OF OPTICS TO PAINTING. 133 
 
 pictures in which the aerial colour, the coloured re- 
 flection and shade, so variously alter the tint of each 
 single coloured surface when it is not perfectly smooth, 
 that it is hardly possible to give an indisputable de- 
 termination of its tint. In such cases, moreover, the 
 direct action of the colour upon the eye is only a 
 subordinate means ; for, on the other hand, the 
 prominent colours and lights must also serve for 
 directing the attention to the more important points 
 of the representation. Compared with these more 
 poetical and psychological elements of the representa- 
 tion, considerations as to the pleasing effect of the 
 colours are thrown into the background. Only in the 
 pure ornamentation on carpets, draperies, ribbons, or 
 architectonic surfaces is there free scope for pure 
 pleasure in the colours, and only there can it develop 
 itself according to its own laws. 
 
 In pictures, too, there is not, as a general rule, 
 perfect equilibrium between the various colours, but 
 one of them preponderates to an extent which corre- 
 sponds to the dominant light. This is occasioned, in 
 the first case, by the truthful imitation of physical 
 circumstances. If the illumination is rich in yellow 
 light, yellow colours will appear brighter and more 
 brilliant than blue ones ; for yellow bodies are those 
 which preferably reflect yellow light ; while that of 
 blue is only feebly reflected, and is mainly absorbed. 
 
134 ON THE RELATION OF OPTICS TO PAINTING. 
 
 Before the shaded parts of blue bodies, the yellow 
 aerial light produces its effect, and imparts to the 
 blue more or less of a grey tint. The same thing 
 happens in front of red and green, though to a less 
 extent, so that, in their shadows, these colours merge 
 into yellow. This also is closely in accordance with 
 the aesthetic requirements of artistic unity of compo- 
 sition in colour. This is caused by the fact that the 
 divergent colours show a relation to the predominant 
 colour, and point to it most distinctly in their shades. 
 Where this is wanting, the various colours are hard 
 and crude ; and, since each one calls attention to itself, 
 they make a motley and disturbing impression ; and, 
 on the other hand, a cold one, for the appearance 
 of a flood of light thrown over the objects is 
 wanting. 
 
 We have a natural type of the harmony which a 
 well-executed illumination of masses of air can produce 
 in a picture, in the light of the setting sun, which 
 throws over the poorest regions a flood of light and 
 colour, and harmoniously brightens them. The 
 natural reason for this increase of aerial illumination 
 lies in the fact, that the lower and more opaque 
 layers of air are in the direction -of the sun, and 
 therefore reflect more powerfully; while at the same 
 time the yellowish red colour of the light which 
 has passed through the atmosphere becomes more dis- 
 
ON THE RELATION OF OPTICS TO PAINTING. 135 
 
 tinct as the length of path increases which it has to 
 traverse, and that further, this coloration is more 
 pronounced as the background falls into shadow. 
 
 In summing up once more these considerations, we 
 have first seen what limitations are imposed on truth 
 to Nature in artistic representation ; how the painter 
 links the principal means which nature furnishes of 
 recognising depths in the field of view, namely binocu- 
 lar vision, which indeed is even turned against him, 
 as it shows unmistakably the flatness of the picture ; 
 how therefore the painter must carefully select, partly 
 the perspective arrangement of his subject, its posi- 
 tion and its aspect, and partly the lighting and 
 shading, in order to give us a directly intelligible 
 image of its magnitude, its shape, and distance, and 
 how a truthful representation of aerial light is one of 
 the most important means of attaining the object. 
 
 We then saw that even the scale of luminous 
 intensity, as met with in the objects, must be trans- 
 formed in the picture to one differing sometimes by a 
 hundredfold; how here, the colour of the object 
 cannot be simply represented by the pigment; that 
 indeed it is necessary to introduce important changes 
 in the distribution of light and dark, of yellowish and 
 of bluish tints. 
 
 The artist cannot transcribe Nature; he must 
 
136 ON THE EELATION OF OPTICS TO PAINTING. 
 
 translate her; yet this translation may give us an 
 impression in the highest degree distinct and forcible, 
 not merely of the objects themselves, but even of the 
 greatly altered intensities of light under which we 
 view them. The altered scale is indeed in many cases 
 advantageous, as it gets rid of everything which, in 
 the actual objects, is too dazzling, and too fatiguing 
 for the eye. Thus the imitation of Nature in the 
 picture is at the same time an ennobling of the im- 
 pression on the senses. In this respect we can often 
 give ourselves up more calmly and continuously, to the' 
 consideration of a work of art, than to that of a real 
 object. The work of art can produce those gradations 
 of light, and those tints in which the modelling of the 
 forms is most distinct and therefore most expressive. 
 It can bring forward a fulness of vivid fervent colours, 
 and by skilful contrast can retain the sensitiveness of 
 the eye in advantageous equilibrium. It can fearlessly 
 apply the entire energy of powerful sensuous impres- 
 sions, and the feeling of delight associated therewith, 
 to direct and enchain the attention ; it can use their 
 variety to heighten the direct understanding of what 
 is represented, and yet keep the eye in a condition of 
 excitation most favourable and agreeable for delicate 
 sensuous impressions. 
 
 If, in these considerations, my having continually 
 laid much weight on the lightest, finest, and most 
 
ON THE EELATION OF OPTICS TO PAINTING. 137 
 
 accurate sensuous intelligibility of artistic representa- 
 tion, may seem to many of you as a very subordinate 
 point a point which, if mentioned at all by writers on 
 aesthetics, is treated as quite accessory I think this 
 is unjustly so. The sensuous distinctness is by no 
 means a low or subordinate element in the action of 
 works of art ; its importance has forced itself the more 
 strongly upon me the more I have sought to discover 
 the physiological elements in their action. 
 
 What effect is to be produced by a work of art, 
 using this word in its highest sense ? It should 
 excite and enchain our attention, arouse in us, in easy 
 play, a host of slumbering conceptions and their cor- 
 responding feelings, and direct them towards a common 
 object, so as to give a vivid perception of all the fea- 
 tures of an ideal type, whose separate fragments lie 
 scattered in our imagination and overgrown by the 
 wild chaos of accident. It seems as if we can only 
 refer the frequent preponderance, in the mind, of art 
 over reality, to the fact that the latter mixes some- 
 thing foreign, disturbing, and even injurious ; while art 
 can collect all the elements for the desired impression, 
 and allow them to act without restraint. The power of 
 this impression will no doubt be greater the deeper, 
 the finer, and the truer to nature is the sensuous 
 impression which is to arouse the series of images 
 and the effects connected therewith. It must act cer- 
 
138 ON THE RELATION OF OPTICS TO PAINTING. 
 
 tainly, rapidly, unequivocably, and with accuracy if it 
 is to produce a vivid and powerful impression. These 
 essentially are the points which I have sought to com- 
 prehend under the name of intelligibility of the work 
 of art. 
 
 Then the peculiarities of the painters' technique 
 (Technik)) to which physiological optical investigation 
 have led us, are often closely connected with the highest 
 problems of art. We may perhaps think that even the 
 last secret of artistic beauty that is; the wondrous 
 pleasure which we feel in its presence is essentially 
 based on the feeling of an easy, harmonic, vivid stream 
 of our conceptions, which, in spite of manifold changes, 
 flow towards a common object, bring to light laws 
 hitherto concealed, and allow us to gaze in the deepest 
 depths of sensation of our own minds. 
 
139 
 
 ^ LIB 11 A K V 
 
 UNIVKRSITY OF Ij 
 
 CAUFOtt* 
 ON THE OEIGIN 
 
 OF THB 
 
 PLANETABY SYSTEM. 
 
 Lecture delivered in Heidelberg and in Cologne, in 1871. 
 
 IT is my intention to bring a subject before you to-day 
 which has been much discussed that is, the hypothesis 
 of Kant and Laplace as to the formation of the celestial 
 bodies, and more especially of our planetary system. 
 The choice of the subject needs no apology. In popular 
 lectures, like the present, the hearers may reasonably 
 expect from the lecturer, that he shall bring before 
 them well-ascertained facts, and the complete results 
 of investigation, and not unripe suppositions, hypothe- 
 ses, or dreams. 
 
 Of all the subjects to which the thought and im- 
 agination of man could turn, the question as to the 
 origin of the world has, since remote antiquity, been 
 the favourite arena of the wildest speculation. Bene- 
 
140 ON THE ORIGIN OF THE PLANETARY SYSTEM. 
 
 ficent and malignant deities, giants, Kronos who 
 devours his children, Niflheim, with the ice-giant 
 Ymir, who is killed by the celestial Asas, 1 that out of 
 him the world may be constructed these are all figures 
 which fill the cosmogonic systems of the more culti- 
 vated of the peoples. But the universality of the fact, 
 that each people develops its own cosmogonies, and 
 sometimes in great detail, is an expression of the 
 interest, felt by all, in knowing what is our own origin, 
 what is the ultimate beginning of the things about 
 us. And with the question of the beginning is 
 closely connected that of the end of all things; for 
 that which may be formed, may also pass away. The 
 question about the end of things is perhaps of greater 
 practical interest than that of the beginning. 
 
 Now, I must premise that the theory which I 
 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 
 fortieth year, we find that they belong mostly to 
 1 Cox's Aryan Mythology, vol. i. 372. Longmans. 
 
ON THE OKIGIN OF THE PLANETARY SYSTEM. 141 
 
 Natural Philosophy, and are far in advance of their 
 times with a number of the happiest ideas. His 
 philosophical writings at this period are but few, and 
 partly like his introductory lecture, directly originat- 
 ing in some adventitious circumstance ; at the sam 
 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 
 inversion of the historical connection, when Kant's 
 
142 ON THE ORIGIN OF THE PLANETARY SYSTEM. 
 
 name is occasionally misused, to recommend that 
 natural philosophy shall leave the inductive method, 
 by which it has become great, to revert to the windy 
 speculations of a so-called 'deductive method.' No 
 one would have attacked such a misuse, more ener- 
 getically and more incisively, than Kant himself 
 if he were still among us. 
 
 The same hypothesis as to the origin of our 
 planetary system was advanced a second time, but 
 apparently quite independently of Kant, by the most 
 celebrated of French astronomers, Simon, Marquis de 
 Laplace. It formed, as it were, the final conclusion of 
 his work on the mechanism of our system, executed 
 with such gigantic industry and great mathematical 
 acuteness. You see from the names of these two men, 
 whom we meet as experienced and tried leaders in our 
 course, that in a view in which they both agree, we 
 have not to deal with a mere random guess, but with a 
 careful and well-considered attempt to deduce conclu- 
 sions as to the unknown past from known conditions of 
 the present time. 
 
 It is in the nature of the case, that a hypothesis as 
 to the origin of the world which we inhabit, and 
 which deals with things in the most distant past, 
 cannot be verified by direct observation. It may, how- 
 ever, receive direct confirmation, if, in the progress of 
 scientific knowledge, new facts accrue to those already 
 
ON THE OEiaiN OF THE PLANETARY SYSTEM. 143 
 
 known, and like them are explained on the hypothesis ; 
 and particularly if survivals of the processes, assumed 
 to have taken place in the formation of the heavenly 
 bodies, can be proved to exist in the present. 
 
 Such direct confirmations of various kinds have, in 
 fact, been formed for the view we are about to discuss, 
 and have materially increased its probability. 
 
 Partly this fact, and partly the fact that the 
 hypothesis in question has recently been mentioned in 
 popular and scientific books, in connection with philo- 
 sophical, ethical, and theological questions, have em- 
 boldened me to speak of it here. I intend not so 
 much to tell you anything substantially new in refer- 
 ence to it, as to endeavour to give, as connectedly as 
 possible, the reasons which have led to, and have 
 confirmed it. 
 
 These apologies which I must premise, only apply 
 to the fact that I treat a theme of this kind as a popular 
 lecture. Science is not only entitled, but is indeed 
 beholden, to make such an investigation. For her it is a 
 definite and important question the question, namely, 
 as to the existence of limits to the validity of the laws 
 of nature, which rule all that now surrounds us ; the 
 question whether they have always held in the past, 
 and whether they will always hold in the future ; or 
 whether, on the supposition of an everlasting unifor- 
 mity of natural laws, our conclusions from present 
 
144 ON THE ORIGIN OF THE PLANETARY SYSTEM. 
 
 circumstances as to the past, and as to the future, 
 imperatively lead to an impossible state of things; 
 that is, to the necessity of an infraction of natural 
 laws, of a beginning which could not have been due 
 to processes known to us. Hence, to begin such an 
 investigation as to the possible or probable primeval 
 history of our present world, is, considered as a ques- 
 tion of science, no idle speculation, but a question as 
 to the limits of its methods, and as to the extent to 
 which existing laws are valid. 
 
 It may perhaps appear rash that we, restricted as 
 we are, in the circle of our observations in space, by our 
 position on this little earth, which is but as a grain of 
 dust in our milky way ; and limited in time by the 
 short duration of the human race ; that we should 
 attempt to apply the laws which we have deduced 
 from the confined circle of facts open to us, to the 
 whole range of infinite space, and of time from 
 everlasting to everlasting. But all our thought and 
 our action, in the greatest as well as in the least, 
 is based on our confidence in the unchangeable order 
 of nature, and this confidence has hitherto been the 
 more justified, the deeper we have penetrated into the 
 interconnections of natural phenomena. And that the 
 general laws, which we have found, also hold for the 
 most distant vistas of space, has acquired strong actual 
 confirmation during the past half-century. 
 
ON THE ORIGIN OF THE PLANETARY SYSTEM. 145 
 
 In the front rank of all, then, is the law of gravita- 
 tion. The celestial bodies, as you all know, float and 
 move in infinite space. Compared with the enormous 
 distances between them, each of us is but as a grain of 
 dust. The nearest fixed stars, viewed even under the 
 most powerful magnification, have no visible diameter ; 
 .and we may be sure that even our sun, looked at from 
 the nearest fixed stars, would only appear as a single 
 luminous point ; seeing that the masses of those stars, 
 in so far as they have been determined, have not been 
 found to be materially different from that of the sun. 
 But, notwithstanding these enormous distances, there 
 is an invisible tie between them which connects them 
 together, and brings them in mutual interdependence. 
 This is the force of gravitation, with which all heavy 
 masses attract each other. We know this force as 
 gravity, when it is operative between an earthly body 
 and the mass of our earth. The force which causes 
 a body to fall to the ground is none other than that 
 which continually compels the moon to accompany the 
 earth in its path round the sun, and which keeps the 
 earth itself from fleeing off into space, away from the 
 sun. 
 
 You may realise, by means of a simple mechanical 
 model, the course of planetary motion. Fasten to the 
 branch of a tree, at a sufficient height, or to a rigid 
 bar, fixed horizontally in the wall, a silk cord, and at 
 
 II. L 
 
146 ON THE ORIGIN OF THE PLANETARY SYSTEM. 
 
 its end a small heavy body for instance, a lead balL 
 If you allow this to hang at rest, it stretches the 
 thread. This is the position of equilibrium of the- 
 ball. To indicate this, and keep it visible, put in 
 the place of the ball any other solid body for in- 
 stance, a large terrestrial globe on a stand. For this 
 purpose the ball must be pushed aside, but it presses 
 against the globe, and, if taken away, it still tends to 
 come back to it, because gravity impels it towards its- 
 position of equilibrium, which is in the centre of the 
 sphere. And upon whatever side it is drawn, the same 
 thing always happens. This force, which drives the 
 ball towards the globe, represents in our model the 
 attraction which the earth exerts on the moon, or the 
 sun on the planets. After you have convinced your- 
 selves of the accuracy of these facts, try to give the 
 ball, when it is a little away from the globe, a slight 
 throw in a lateral direction. If you have accurately 
 hit the strength of the throw, the small ball will 
 move round the large one in a circular path, and may 
 retain this motion for some time ; just as the moon 
 persists in its course round the earth, or the planets 
 about the sun. Now, in our model, the circles 
 described by the lead ball will be continually narrower,, 
 because the opposing forces, the resistance of the air,, 
 the rigidity of the thread, friction, cannot be elimi- 
 nated, in this case, as they are excluded in the plane- 
 tary system. 
 
ON THE ORIGIN OF THE PLANETAEY SYSTEM. 147 
 
 If the path about the attracting centre is exactly 
 circular, the attracting force always acts on the 
 planets, or on the lead sphere, with equal strength. 
 In this case, it is immaterial according to what law 
 the force would increase or diminish 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 
 
 FIG. />. 
 
 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 ha* 
 such a position in reference to the attracting centre,, 
 that this is in the centre, c, of the ellipse. For planets, 
 on the contrary, the attracting force is feebler the 
 
 L 2 
 
148 ON THE ORIGIN OF THE PLANETARY SYSTEM. 
 
 further it is removed from the attracting body, and 
 this is the reason that an ellipse is described, one of 
 whose foci lies in the centre of attraction. The two 
 foci, a and 6, are two points which lie symmetrically 
 towards the ends of the ellipse, and are characterised 
 by the property that the sum of their distances, 
 am+bm, is the same from any given points. 
 
 Kepler had found that the paths of the planets are 
 ellipses of this kind ; and since, as the above example 
 shows, the form and position of the orbit depend on 
 the law according to which the magnitude of the 
 attracting force alters, Newton could deduce from the 
 form of the planetary orbits the well-known law of the 
 force of gravitation, which attracts the planets to the 
 sun, according to which this force decreases with 
 increase of distance as the square of that distance. 
 Terrestrial gravity must obey this law, and Newton 
 had the wonderful self-denial to refrain from publish- 
 ing his important discovery until it had acquired a 
 direct confirmation; this followed from the observa- 
 tions, that the force which attracts the moon towards 
 the earth, bears towards the gravity of a terrestrial 
 body the ratio required by the above law. 
 
 In the course of the eighteenth century the power 
 of mathematical analysis, and the methods of astrono- 
 mical observation, increased so far that all the compli- 
 cated actions, which take place between all the planets, 
 
ON THE ORIGIN OF THE PLANETARY SYSTEM. 149 
 
 and all their satellites, in consequence of the mutual 
 action of each upon each, and which astronomers call 
 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 
 Bessel's calculation of the discrepancy between the 
 actual and the calculated motion of Uranus, it was 
 inferred that there must be another planet. The 
 position of this planet was calculated by Leverrier and 
 Adams, and thus Neptune, the most distant of all 
 known at that time, was discovered. 
 
 But it was not merely in the region of the attrac- 
 tion of our sun that the law of gravitation was found 
 to hold. With regard to the fixed stars, it was found 
 
150 ON THE ORIGIN OF THE PLANETARY SYSTEM. 
 
 that double stars moved about each other in elliptical 
 paths, and that therefore the same law of gravitation 
 must hold for them as for our planetary system. The 
 distance of some of them could be calculated. The 
 nearest of them, a, in the constellation of the Centaur, 
 is 1,039,600 miles 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 three 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 traverse ; as, for instance, the Pole Star ; 
 but the law of gravitation is seen to hold, ruling the 
 motion of the double stars, at distances in the heavens, 
 which all the means we possess have hitherto utterly 
 failed to measure. 
 
 The knowledge of the law of gravitation has here 
 also led to the discovery of new bodies, as in the 
 case of Neptune. Peters of Altona found, confirming 
 therein a conjecture of Bessel, that Sirius, the most 
 brilliant of the fixed stars, moves in an elliptical path 
 about an invisible centre. This must have been due 
 to an unseen companion, and when the excellent and 
 powerful telescope of the University of Cambridge, in 
 the United States, had been set up, this was discovered. 
 It is not quite dark, but its light is so feeble that it 
 
ON THE ORIGIN OF THE PLANETARY SYSTEM. 151 
 
 can only be seen by the most perfect instruments. 
 The mass of Sirius is found to be 13-76, and that of 
 its satellite 6'7l, times the mass of the sun; their 
 mutual distance is equal to thirty-seven times the 
 radius of the earth's orbit, and is therefore somewhat 
 larger than the distance of Neptune from the sun. 
 
 Another fixed star, Procyon, is in the same case as 
 Sirius, but its satellite has not yet been discovered. 
 
 You thus see that in gravitation we have dis- 
 covered a property common to all matter, which is not 
 confined to bodies in our system, but extends, as far in 
 the celestial space, as our means of observation have 
 hitherto been able to penetrate. 
 
 But not merely is this universal property of all 
 mass shared by the most distant celestial bodies, as 
 well as by terrestrial ones ; but spectrum analysis has 
 taught us that a number of well-known terrestrial 
 elements are met with in the atmospheres of the 
 fixed stars, and even of the nebulae. 
 
 You all know that a fine bright line of light, seen 
 through a glass prism, appears as a coloured band, red 
 and yellow at one edge, blue and violet at the other, 
 and green in the middle. Such a coloured image is 
 called a spectrum the rainbow is such a one, produced 
 by the refraction of light, though not exactly by a 
 prism ; and it exhibits therefore the series of colours 
 into which white sunlight can thus be decomposed. 
 
152 ON THE OKIG-IN OF THE PLANETARY SYSTEM. 
 
 The formation of the prismatic spectrum depends on 
 the fact that the sun's light, and that of most ignited 
 bodies, is made up of various kinds of light, which 
 appear of different colours to our eyes, and the rays 
 of which are separated from each other when refracted 
 by a prism. 
 
 Now if a solid or a liquid is heated to such an 
 extent that it becomes incandescent, the spectrum 
 which its light gives is, like the rainbow, a broad 
 coloured band without any breaks, with the well-known 
 series of colours, red, yellow, green, blue, and violet,, 
 and in no wise characteristic of the nature of the body 
 which emits the light. 
 
 The case is different if the light is emitted by an 
 ignited gas, or by an ignited vapour that is, a sub- 
 stance vaporised by heat. The spectrum of such a 
 body consists, then, of one or more, and sometimes 
 even a great number, of entirely distinct bright lines, 
 whose position and arrangement in the spectrum is 
 characteristic for the substances of which the gas or 
 vapour consists, so that it can be ascertained, by means 
 of spectrum analysis, what is the chemical constitution 
 of the ignited gaseous body. Graseous spectra of this 
 kind are shown in the heavenly space by many 
 nebulae; for the most part they are spectra which 
 show the bright line of ignited hydrogen and oxygen, 
 and along with it a line which, as yet, has never been 
 
ON THE ORIGIN OF THE PLANETARY SYSTEM. 153 
 
 again found in the spectrum of any terrestrial element. 
 Apart from the proof of two well-known terrestrial 
 elements, this discovery was of the utmost importance,, 
 since it furnished the first unmistakable proof that the 
 cosmical nebulae are not, for the most part, small heaps- 
 of fine stars, but that the greater part of the light 
 which they emit is really due to gaseous bodies. 
 
 The gaseous spectra present a different appearance 
 when the gas is in front of an ignited solid whose 
 temperature is far higher than that of the gas. The 
 observer sees then a continuous spectrum of a solid, 
 but traversed by fine dark lines, which are just visible 
 in the places in which the gas alone, seen in front of 
 a dark background, would show bright lines. The 
 solar spectrum is of this kind, and also that of a great 
 number of fixed stars. The dark lines of the solar 
 spectrum, originally discovered by Wollaston, were 
 first investigated and measured by Fraunhofer, and are 
 hence known as Fraunhofer's lines. 
 
 Far more powerful apparatus was afterwards used 
 by Kirchhoff, and then by Angstrom, to push the de- 
 composition of light as far as possible. Fig. 6 re- 
 presents an apparatus with four prisms, constructed 
 by Steinheil for Kirchhoff. At the further end of the 
 telescope B is a screen with a fine slit, represent- 
 ing a fine slice of light, which can be narrowed or 
 widened by the small screw, and by which the light 
 
154 ON THE ORIGIN OF THE PLANETARY SYSTEM. 
 
 under investigation can be allowed to enter. It then 
 passes through the telescope B, afterwards through the 
 
 FIG. 6. 
 
 four prisms, and finally through the telescope A, from 
 which it reaches the eye of the observer. Figs. 7, 8, 
 FIG. 7. FIG. 8. 
 
ON THE OKIGIN OF THE PLANETARY SYSTEM. 155 
 
 and 9 represent small portions of the solar spectrum 
 as mapped by Kirchoff, taken from the green, yellow, 
 and golden-yellow, in which the chemical symbols 
 below Fe (iron), Ca (calcium), Na (sodium), Pb (lead) 
 and the affixed lines, indicate the positions in which 
 the vapours of these metals, when made incandescent, 
 either in the flames or in the electrical spark, would 
 
 FIG. 9. 
 
 show bright lines. The numbers above them show 
 how far these fractions of Kirchhoff's map of the whole 
 system are apart from each other. Here, also, we see 
 a predominance of iron lines. In the whole spectrum 
 Kirchhoff found not less than 450. 
 
 It follows from this, that the solar atmosphere con- 
 tains an abundance of the vapours of iron, which, by 
 the way, justifies us in concluding what an enormously 
 high temperature must prevail there. It shows, more- 
 
156 ON THE OEIGIN OF THE PLANETAKY SYSTEM. 
 
 over, how our figs. 7, 8, and 9 indicate iron, calcium., 
 and sodium, and also the presence of hydrogen, of zinc,, 
 of copper, and of the metals of magnesia, alumina, 
 baryta, and other terrestrial elements. Lead, on the 
 other hand, is wanting, as well as gold, silver, mercury, 
 antimony, arsenic, and some others. 
 
 The spectra of several fixed stars are similarly con- 
 stituted ; they show systems of fine lines which can be 
 identified with those of terrestrial elements. In the 
 atmosphere of Aldebaran in Taurus there is, again, 
 hydrogen, iron, magnesium, calcium, sodium, and also 
 mercury, antimony, and bismuth ; and, according to 
 H. 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 to substances unknown to us, it is also 
 possible that they are produced by the excessively high 
 temperature of the sun, far transcending anything we 
 can produce. But this is certain, that the known 
 terrestrial substances are widely diffused in space, and 
 especially nitrogen, which constitutes the greater part 
 
ON THE ORIGIN OF THE PLANETARY SYSTEM. 157 
 
 of our atmosphere, and hydrogen, an element in water, 
 which indeed is formed by its combustion. Both have 
 been found in the irresolvable nebulae, and, from the 
 inalterability of their shape, these must be masses 
 of enormous dimensions and at an enormous distance. 
 For this reason Sir W. Herschel considered that they 
 did not belong to the system of our fixed stars, but 
 were representatives of the manner in which other 
 systems manifested themselves. 
 
 Spectrum analysis has further taught us more 
 about the sun, by which he is brought nearer to us, as it 
 were, than could formerly have seemed possible. You 
 know that the sun is an enormous sphere, whose 
 diameter is 112 times as great as that of the earth. 
 We may consider what we see on its surface as a layer 
 of incandescent vapour, which, to judge from the 
 appearances of the sun-spots, has a depth of about 
 500 miles. This layer of vapour, which is continually 
 radiating heat on the outside, and is certainly cooler 
 than the inner masses of the sun, is, however, hotter 
 than all our terrestrial flames hotter even than the 
 incandescent carbon points of the electrical arc, which 
 represent the highest temperature attainable by terres- 
 trial means. This can be deduced with certainty from 
 Kirchhoff's law of the radiation of opaque bodies, from 
 the greater luminous intensity of the sun. The older 
 assumption, that the sun is a dark cool body, sur- 
 
158 ON THE ORIGIN OF THE PLANETARY SYSTEM. 
 
 rounded by a photosphere which only radiates heat 
 and light externally, contains a physical impossibility. 
 
 Outside the opaque photosphere, the sun appears 
 surrounded by a layer of transparent gases, which are 
 hot enough to show in the spectrum bright coloured 
 lines, and are hence called the Chromosphere. They 
 show the bright lines of hydrogen, of sodium, of magne- 
 sium, and iron. In these layers of gas and of vapour 
 about the sun enormous storms occur, which are as 
 much greater than those of our earth in extent and in 
 velocity as the sun is greater than the earth. Currents 
 of ignited hydrogen burst out several thousands of miles 
 high, like gigantic jets or tongues of flame, with clouds 
 of smoke 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. We now possess a method, devised by 
 MM. Jansen and Lockyer, by which they may at any 
 time be seen by the aid of the spectroscope. 
 
 On the other hand, there are individual darker 
 parts on the sun's surface, what are called sun-spots, 
 which were seen as long ago as by Galileo. They are 
 funnel-shaped, the sides of the funnel are not so dark 
 as the deepest part, the core. Fig. 10 represents such 
 
 1 According to H. C. Vogel's observations in Bothkamp to a height 
 of 70,000 miles. The spectroscopic displacement of the lines showed 
 velocities of 18 to 23 miles in a second; and, according to Lockyer, of 
 even 37 to 42 miles. 
 
ON THE ORIGIN OF THE PLANETARY SYSTEM. 159 1 
 
 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 near the edge, and are therefore seen in a very 
 
160 ON THE OKIGIN OF THE PLANETARY SYSTEM. 
 
 oblique direction. Fig. 11 represents in A to E the 
 different aspects of such a spot as it comes near the 
 edge of the sun. 
 
 Just on the edge of these spots there are spectro- 
 scopic indications of the most violent motion, and 
 in their vicinity there are often large protuberances ; 
 they show comparatively often a rotatory motion. 
 They may be considered to be places where the 
 FIG. 11. 
 
 H) 
 
 cooler gases from the outer layers of the sun's atmos- 
 phere sink down, and perhaps produce local superficial 
 coolings of the sun's mass. To understand the origin 
 of these phenomena, it must be remembered that the 
 gases, as they rise from the hot body of the sun, are 
 charged with vapours of difficultly volatile metals, 
 which expand as they ascend, and partly by their ex- 
 pansion, and partly by radiation into space, must be- 
 come cooled. At the same time, they deposit their 
 
ON THE ORIGIN OF THE PLANET AEY SYSTEM. 161 
 
 more difficultly volatile constituents as fog or cloud. 
 This cooling can only, of course, be regarded as com- 
 parative ; their temperature is probably, even then, 
 higher than any temperature attainable on the earth. 
 If now the upper layers, freed from the heavier 
 vapours, sink down, there will be a space over the 
 sun's body which is free from cloud. They appear 
 then as depressions, because about them are layers of 
 ignited vapours as much as 500 miles in height. 
 
 Violent storms cannot fail to occur in the sun's 
 atmosphere, because it is cooled on the outside, and 
 the coolest and comparatively densest and heaviest 
 parts come to lie over the hotter and lighter ones. 
 This is the reason why we have frequent, and at times 
 sudden and violent, movements in the earth's atmos- 
 phere, because this is heated from the ground made 
 hot by the sun and is cooled above. With the far 
 more colossal magnitude and temperature of the sun, 
 its meteorological processes are on a far larger scale, 
 and are far more violent. 
 
 We will now pass to the question of the perman- 
 ence of the present condition of our system. For a 
 long time the view was pretty generally held that, in 
 its chief features at any rate, it was unchangeable. 
 This opinion was based mainly on the conclusions at 
 which Laplace had arrived as the final results of his 
 long and laborious investigations, of the influence of 
 n. M 
 
162 ON THE OKKHN OF THE PLANETARY SYSTEM. 
 
 planetary disturbances. By disturbances of the plane- 
 tary motion astronomers understand, as I have already 
 mentioned, those deviations from the purely elliptical 
 motion which are due to the attraction of various planets 
 and satellites upon each other. The attraction of the 
 sun, as by far the largest body of our system, is indeed 
 the chief and preponderating force which produces the 
 motion of the planets. If it alone were operative, 
 each of the planets would move continuously in a con- 
 stant ellipse whose axes would retain the same direc- 
 tion and the same magnitude, making the revolutions 
 always in the same length of time. But, in point of 
 fact, in addition to the attraction of the sun there are 
 the attractions of all other planets, which, though 
 small, yet, in long periods of time, do effect slow 
 changes in the plane, the direction, and the magnitude 
 of the axes of its elliptical orbit. It has been asked 
 whether these attractions in the orbit of the planet 
 could go so far as to cause two adjacent planets to 
 encounter each other, so that individual ones fall into 
 the sun. Laplace was able to reply that this could not 
 be the case ; that all alterations in the planetary orbits 
 produced by this kind of disturbance must periodically 
 increase and decrease, and again revert to a mean 
 condition. But it must not be forgotten that this 
 result of Laplace's investigations only applies to dis- 
 turbances due to the reciprocal attraction of planets 
 
ON THE ORIGIN OF THE PLANETARY SYSTEM. 163 
 
 upon each other, and on the assumption that no forces 
 of other kinds have any influence on their motions. 
 
 On our earth we cannot produce such an everlast- 
 ing motion as that of the planets seems to be; for 
 resisting forces are continually being opposed to all 
 movements of terrestrial bodies. The best known of 
 these are what we call friction, resistance of the air, 
 and inelastic impact. 
 
 Hence the fundamental law of mechanics, accord- 
 ing to which every motion of a body on which no force 
 acts goes on in a straight line for ever with unchanged 
 velocity, never holds fully. 
 
 Even if we eliminate the influence of gravity in 
 a ball, for example, which rolls on a plane surface, we 
 see it go on for a while, and the further the smoother 
 is the path ; but at the same time we hear the rolling 
 ball make a clattering sound that is, it produces waves 
 of sound in the surrounding bodies; there is friction 
 even on the smoothest surface ; this sets the surround- 
 ing air in vibration, and imparts to it some of its own 
 motion. Thus it happens that its velocity is con- 
 tinually less and less until it finally ceases. In like 
 manner, even the most carefully constructed wheel 
 which plays upon fine points, once made to turn, goes 
 on for a quarter of an hour, or even more, but then 
 stops. For there is always some friction on the axles, 
 and in addition there is the resistance of the air, which 
 
 M 2 
 
164 ON THE ORIG-IN OF THE PLANETAEY SYSTEM. 
 
 resistance is mainly due to that of the particles of air 
 against each other, due to their friction against the 
 wheel. 
 
 If we could once set a body in rotation, and keep 
 it from falling, without its being supported by another 
 body, and if we could transfer the whole arrangement to 
 an absolute vacuum, it would continue to move for ever 
 with undiminished velocity. This case, which cannot 
 be realised on terrestrial bodies, is apparently met 
 with in the planets with their satellites. They appear 
 to move in the perfectly vacuous cosmical space, with- 
 out contact with any body which could produce 
 friction, and hence their motion seems to be one which 
 never diminishes. 
 
 You see, however, that the justification of this 
 conclusion depends on the question whether cosmical 
 space is really quite vacuous. Is there nowhere any 
 friction in the motion of the planets ? 
 
 From the progress which the knowledge of nature 
 has made since the time of Laplace, we must now 
 answer both questions in the negative. 
 
 Celestial space is not absolutely vacuous. In the 
 first place, it is filled by that continuous medium the 
 agitation of which constitutes light and radiant heat, 
 and which physicists know as the luminiferous ether. 
 In the second place, large and small fragments of 
 heavy matter, from the size of huge stones to that of 
 
ON THE OKiaiN OF THE PLANETARY SYSTEM. 165 
 
 dust, are still everywhere scattered; at any rate, in 
 those parts of space which our earth traverses. 
 
 The existence of the luminiferous ether cannot be 
 considered doubtful. That light and radiant heat are 
 due to a motion which spreads in. all directions has been 
 sufficiently proved. For the transference of such a 
 motion through space there must be something which 
 can be moved. Indeed, from the magnitude of the 
 action of this motion, or from that which the science 
 of mechanics calls its vis viva, we may indeed assign 
 certain limits for the density of this medium. Such 
 a calculation has been made by Sir W. Thomson, the 
 celebrated Glasgow physicist. He has found that the 
 density may possibly be far less than that of the air 
 in the most perfect exhaustion obtainable by a good 
 air-pump ; but that the mass of the ether cannot be 
 absolutely equal to zero. A volume equal to that of 
 the earth cannot contain less than 2,775 pounds of 
 luminous ether. 1 
 
 The phenomena in celestial space are in conformity 
 with this. Just as a heavy stone flung through the 
 air shows scarcely any influence of the resistance of 
 the air, while a light feather is appreciably hindered ; 
 in like manner the medium which fills space is far too 
 attenuated for any diminution to have been perceived 
 
 1 This calculation would, however, lose its bases if Maxwell's hypo- 
 thesis were confirmed, according to which light depends on electrical 
 and magnetical oscillations 
 
166 ON THE OEiaiN OF THE PLANETARY SYSTEM. 
 
 in the motion of the planets since the time in which 
 we possess astronomical observations of their path. It 
 is different with the smaller bodies of our system. 
 Encke in particular has shown, with reference to the 
 well-known small comet which bears his name, that it 
 circulates round the sun in ever-diminishing orbits and 
 in ever shorter periods of revolution. Its motion is 
 similar to that of the circular pendulum which we have 
 mentioned, and which, having its velocity gradually 
 delayed by the resistance of the air, describes circles 
 about its centre of attraction, which continually become 
 smaller and smaller. The reason for this phenomenon 
 is the following: The force which offers a resistance 
 to the attraction of the sun on all comets and planets, 
 and which prevents them from getting continually 
 nearer to the sun, is what is called the centrifugal 
 force that is, the tendency to continue their motion 
 in a straight line in the direction of their path. As 
 the force of their motion diminishes, they yield by a 
 corresponding amount to the attraction of the sun, and 
 get nearer to it. If the resistance continues, they will 
 continue to get nearer the sun until they fall into it. 
 Encke's comet is no doubt in this condition. But the 
 resistance whose presence in space is hereby indicated, 
 must act, and has long continued to act, in the same 
 manner on the far larger masses of the planets. 
 
 The presence of partly fine and partly coarse 
 
ON THE ORIGIN OF THE PLANETARY SYSTEM. 167 
 
 heavy masses diffused in cosmical space is more dis- 
 tinctly revealed by the phenomena of asteroids and of 
 meteorites. We know now that these are bodies 
 which ranged about in cosmical space, before they came 
 within the region of our terrestrial atmosphere. In 
 the more strongly resisting medium which this atmos- 
 phere offers they are delayed in their motion, and at 
 the same time are heated by the corresponding friction. 
 Many of them may still find an escape from the terres- 
 trial atmosphere, and continue their path through 
 space with an altered and retarded motion. Others 
 fall to the earth ; the larger ones as meteorites, while 
 the smaller ones are probably resolved into dust by the 
 heat, and as such fall without being seen. According 
 to Alexander Herschel's estimate, we may figure shoot- 
 ing-stars as being on an average of the same size as 
 paving-stones. Their incandescence mostly occurs in 
 the higher and most attenuated regions of the atmos- 
 phere, eighteen miles and more above the surface of 
 the earth. As they move in space under the influence 
 of the same laws as the planets and comets, they 
 possess a planetary velocity of from eighteen to forty 
 miles in a second. By this, also, we observe that they 
 are in fact stelle cadente, falling stars, as they have 
 long been called by poets. 
 
 This enormous velocity with which they enter our 
 atmosphere is undoubtedly the cause of their becom- 
 
168 ON THE OKIGIN OF THE PLANETAEY SYSTEM. 
 
 ing heated. You all know that friction heats the 
 bodies rubbed. Every match that we ignite, every 
 badly greased coach-wheel, every auger which we work 
 in hard wood, teaches this. The air, like solid bodies, 
 not only becomes heated by friction, but also by the 
 work consumed in its compression. One of the most 
 important results of modern physics, the actual proof 
 of which is mainly due to the Englishman Joule, is 
 that, in such a case, the heat developed is exactly pro- 
 portional to the work expended. If, like the mechani- 
 cians, we measure the work done by the weight which 
 would be necessary to produce it, multiplied by the 
 height from which it must fall, Joule has shown that 
 the work, produced by a given weight of water falling 
 through a height of 425 metres, would be just suffi- 
 cient to raise the same weight of water through one 
 degree Centigrade. The equivalent in work of a 
 velocity of eighteen to twenty-four miles in a second 
 may be easily calculated from known mechanical laws ; 
 and this, transformed into heat, would be sufficient to 
 raise the temperature of a piece of meteoric iron to 
 900,000 to 2,500,000 degrees Centigrade, provided that 
 all the heat were retained by the iron, and did not, as 
 it undoubtedly does, mainly pass into the air. This 
 calculation shows, at any rate, that the velocity of the 
 shooting-stars is perfectly adequate to raise them to 
 the most violent incandescence. The temperatures 
 
ON THE OKIGIN OF THE PLANET AEY SYSTEM. 169 
 
 attainable by terrestrial means scarcely exceed 2,000 
 degrees. In fact, the outer crusts of meteoric stones 
 generally show traces of incipient fusion ; and in cases 
 in which observers examined with sufficient prompti- 
 tude the stones which had fallen they found them hot 
 on the surface, while the interior of detached pieces 
 seemed to show the intense cold of cosmical space. 
 
 To the individual observer who casually looks 
 towards the starry sky the meteorites appear as a rare 
 and exceptional phenomenon. If, however, they are 
 continuously observed, they are seen with tolerable 
 regularity, especially towards morning, when they 
 usually fall. But a single observer only views but a 
 small part of the atmosphere; and if they are calcu- 
 lated for the entire surface of the earth it results that 
 about seven and a half millions fall every day. In our 
 regions of space, they are somewhat sparse and distant 
 from each other. According to Alexander Herschel's 
 estimates, each stone is, on an average, at a distance of 
 450 miles from its neighbours. But the earth moves 
 through 18 miles every second, and has a diameter of 
 7,820 miles, and therefore sweeps through 876 millions 
 of cubic miles of space every second, and carries with 
 it whatever stones are contained therein. 
 
 Many groups are irregularly distributed in space, 
 being probably those which have already undergone 
 disturbances by planets. There are also denser swarms 
 
170 ON THE ORIGIN OF THE PLANETARY SYSTEM. 
 
 which move in regular elliptical orbits, cutting the 
 earth's orbit in definite places, and therefore always 
 oc$ur 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, have 
 perhaps been formed by the gradual aggregation of 
 
^ / 
 
 ON THE ORIGIN OF THE PLANETARY SYSTEM, 
 
 ' 
 
 fine dust, we see ourselves led to a pfri 
 
 ~*+ / 
 
 of fine nebulous masses. () 
 
 - 
 
 From this point of view, that the fall of shooti 
 stars and of meteorites is perhaps only a small survival < 
 of a process which once built up worlds, it assumes far 
 greater significance. 
 
 This would be a supposition of which we might 
 admit the possibility, but which could not perhaps 
 claim any great degree of probability, if we did not 
 find that our predecessors, starting from quite different 
 considerations, had arrived at the same hypothesis. 
 
 You know that a considerable number of planets 
 rotate around the sun besides the eight larger ones, 
 Mercury, Venus, the Earth, Mars, Jupiter, Saturn, 
 Uranus, and Neptune; in the interval between Mars 
 and Jupiter there circulate, as far as we know, 156 
 small planets or planetoids. Moons also rotate about 
 the larger planets that is, about the Earth and the 
 four most distant ones, Jupiter, Saturn, Uranus, and 
 Neptune; and lastly the Sun, and at any rate the 
 larger planets, rotate about their own axes. Now, in 
 the first place, it is remarkable that all the planes of 
 rotation of the planets and of their satellites, as well 
 as the equatorial planes of these planets, do not vary 
 much from each other, and that in these planes all the 
 rotation is in the same direction. The only consider- 
 able exceptions known are the moons of Uranus, 
 
172 ON THE OKIGIN OF THE PLANETAEY SYSTEM. 
 
 whose plane is almost at right angles 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 circular 
 
ON THE ORIGIN OF THE PLANETAEY SYSTEM. 173 
 
 ones which we meet with in all the bodies we have 
 named. There is, in fact, a complete irregularity in 
 the comets and meteoric swarms, which we have 
 much reason for considering to be formations which 
 have only accidentally come within the sphere of the 
 sun's attraction. 
 
 The number of coincidences in the orbits of the 
 planets and their satellites is too great to be ascribed 
 to accident. We must inquire for the reason of this 
 coincidence, and this can only be sought in a primi- 
 tive connection of the entire mass. Now, we are 
 acquainted with forces and processes which condense 
 an originally diffused mass, but none which could drive 
 into space such large masses, as the planets, in the 
 condition we now find them. Moreover, if they had 
 become detached from the common mass, at a place 
 much nearer the sun, they ought to have a markedly 
 elliptical orbit. We must assume, accordingly, that 
 this mass in its primitive condition extended at least 
 to the orbit of the outermost planets. 
 
 These were the essential features of the considera- 
 tions which led Kant and Laplace to their hypothesis. 
 In their view our system was originally a chaotic ball 
 of nebulous matter, of which originally, when it ex- 
 tended to the .path of the most distant planet, many 
 billions of cubic miles could contain scarcely a gramme 
 of mass. This ball, when it had become detached from 
 
174 ON THE ORIGIN OF THE PLANETABY SYSTEM. 
 
 the nebulous balls of the adjacent fixed stars, possessed 
 a slow movement of rotation. It became condensed 
 under the influence of the reciprocal attraction of its 
 parts ; and, in the degree in which it condensed, the 
 rotatory motion increased, and formed it into a flat 
 disk. From time to time masses at the circumference 
 of this disk became detached under the influence of 
 the increasing centrifugal force ; that which became 
 detached formed again into a rotating nebulous mass, 
 which either simply condensed and formed a planet, or 
 during this condensation again repelled masses from 
 the periphery, which became satellites, or in one case, 
 that of Saturn, remained as a coherent ring. In an- 
 other case, the mass which separated from the outside 
 of the chief ball, divided into many parts, detached 
 from each other, and furnished the swarms of small 
 planets between Mars and Jupiter. 
 
 Our more recent experience as to the nature of 
 star showers teaches us that this process of the conden- 
 sation of loosely diffused masses to form larger bodies 
 is by no means complete, but still goes on, though the 
 traces are slight. The form in which it now appears 
 is altered by the fact that meanwhile the gaseous 
 or dust-like mass diffused in space had united under 
 the influence of the force of attraction, and of the 
 force of crystallisation of their constituents, to larger 
 pieces than originally existed. 
 
ON THE ORIGIN OF THE PLANETARY SYSTEM. 175 
 
 The showers of stars, as examples now taking place 
 of the process which formed the heavenly bodies, are 
 important from another point of view. They develop 
 light and heat; and that directs us to a third series 
 of considerations, which leads again to the same goal. 
 
 All life and all motion on our earth is, with few 
 exceptions, kept up by a single force, that of the sun's 
 rays, which bring to us light and heat. They warm 
 the air of the hot zones, this becomes lighter and 
 ascends, while the colder air flows towards the poles. 
 Thus is formed the great circulation of the passage- 
 winds. Local differences of temperature over land and 
 sea, plains and mountains, disturb the uniformity of 
 this great motion, and produce for us the capricious 
 change of winds. Warm aqueous vapours ascend with 
 the warm air, become condensed into clouds, and fall 
 in the cooler zones, and upon the snowy tops of the 
 mountains, as rain and as snow. The water collects in 
 brooks, in rivers, moistens the plains, and makes life 
 possible ; crumbles the stones, carries their fragments 
 along, and thus works at the geological transformation 
 of the earth's surface. It is only under the influence 
 of the sun's rays that the variegated covering of plants 
 of the earth grows ; and while they grow, they accumu- 
 late in their structure organic matter, which partly 
 serves the whole animal kingdom as food, and serves 
 man more particularly as fuel. Coals and lignites, the 
 
176 ON THE ORIGIN OF THE PLANETARY SYSTEM. 
 
 sources of power of our steam engines, are remains of 
 primitive plants the ancient production of the sun's 
 rays. 
 
 Need we wonder if, to our forefathers of the Aryan 
 race in India and Persia, the sun appeared as the fittest 
 symbol of the Deity ? They were right in regarding it 
 as the giver of all life as the ultimate source of almost 
 all that has happened on earth. 
 
 But whence does the sun acquire this force? It 
 radiates forth a more intense light than can be attained 
 with any terrestrial means. It yields as much heat as 
 if 1,500 pounds of coal were burned every hour upon 
 each square foot of its surface. Of the heat which 
 thus issues from it, the small fraction which enters our 
 atmosphere furnishes a great mechanical force. Every 
 steam-engine teaches us that heat can produce such 
 force. The sun, in fact, drives on earth a kind of 
 steam-engine whose performances are far greater than 
 those of artificially constructed machines. The circu- 
 lation of water in the atmosphere raises, as has been 
 said, the water evaporated from the warm tropical 
 seas to the mountain heights ; it is, as it were, a water- 
 raising engine of the most magnificent kind, with 
 whose power no artificial machine can be even dis- 
 tantly compared. I have previously explained the 
 mechanical equivalent of heat. Calculated by that 
 standard, the work which the sun produces by its 
 
ON THE ORIOIN OF THE PLANETARY SYSTEM. 177 
 
 radiation is equal to the constant exertion of 7,000 
 horse-power for each square foot of the sun's surface. 
 
 For a long time experience had impressed on our 
 mechanicians that a working force cannot be produced 
 from nothing; that it can only be taken from the 
 stores which nature possesses ; which are strictly limited 
 and which cannot be increased at pleasure whether it 
 be taken from the rushing water or from the wind ; 
 whether from the layers of coal, or from men and from 
 animals, which cannot work without the consumption 
 of food. Modern physics has attempted to prove the 
 universality of this experience, to show that it applies 
 to the great whole of all natural processes, and is inde- 
 pendent of the special interests of man. These have 
 been generalised and comprehended in the all-ruling 
 natural law of the Cons elevation 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 like 
 
 II. N 
 
178 ON THE OKIGIN OF THE PLANETARY SYSTEM. 
 
 matter itself. It seems as if Ofoethe 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 how know that the sun contains the terrestrial 
 elements with which we are acquainted. Let 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 
 would unite to form water. The mass of the sun is 
 
ON THE ORIGIN OF THE PLANETARY SYSTEM. 179 
 
 known, and also the quantity of heat produced by the 
 union of known weights of oxygen and hydrogen. 
 Calculation shows that under the above supposition, 
 the heat resulting from their combustion would be 
 sufficient to keep up the radiation of heat from the 
 sun for 3,021 years. That, it is true, is a long time, 
 but even profane history teaches that the sun has 
 lighted and warmed us for 3,000 years, and geology 
 puts it beyond doubt that this period must be ex- 
 tended to millions of years. 
 
 Known chemical forces are thus so completely in- 
 adequate, even on the most favourable assumption, to 
 explain the production of heat which takes place in 
 the sun, that we must' quite drop this hypothesis. 
 
 We must seek for forces of far greater magnitude, 
 and these we can only find in cosmical attraction. We 
 have already seen that the comparatively small masses 
 of shooting-stars and meteorites can produce extra- 
 ordinarily large amounts of heat when their cosmical 
 velocities are arrested by our atmosphere. Now the 
 force which has produced these great velocities is 
 gravitation. We know of this force as one acting on 
 the surface of our planet when it appears as terrestrial 
 gravity. We know that a weight raised from the 
 earth can drive our clocks, and that in like manner 
 the gravity of the water rushing down from the moun- 
 tains works our mills. 
 
 N 2 
 
180 ON THE OEIGIN OF THE PLANETAEY SYSTEM. 
 
 If a weight falls from a height and strikes the 
 ground its mass loses, indeed, the visible motion which 
 it had as a whole in fact, however, this motion is not 
 lost; it is transferred to the smallest elementary 
 particles of the mass, and this invisible vibration of 
 the molecules is the motion of heat. Visible motion 
 is transformed by impact, into the motion of heat. 
 
 That which holds in this respect for gravity, holds 
 also for gravitation, A heavy mass, of whatever kind, 
 which is suspended in space separated from another 
 heavy mass, represents a force capable of work. For 
 both masses attract each other, and, if unrestrained by 
 centrifugal force, they move towards each other under 
 the influence of this attraction ; this takes place with 
 ever-increasing velocity ; and if this velocity is finally 
 destroyed, whether this be suddenly, by collision, or 
 gradually, by the friction of movable parts, it develops 
 the corresponding quantity of the motion of heat, the 
 amount of which can be calculated from the equiva- 
 lence, previously established, between heat and me- 
 chanical work. 
 
 Now we may assume with great probability that 
 very many more meteors fall upon the sun than upon 
 the earth, and with greater velocity, too, and therefore 
 give more heat. Yet the hypothesis, that the entire 
 amount of the sun's heat which is continually lost by 
 radiation, is made up by the 'fall of meteors, a hypothesis 
 
ON THE OKIGIN OF THE PLANETAKY SYSTEM. 181 
 
 which was propounded by Mayer, and has been favour- 
 ably adopted by several other physicists, is open, ac- 
 cording to Sir W. Thomson's investigations, to ob- 
 jection ; for, assuming it to hold, the mass of the 
 sun should increase so rapidly that the consequences 
 would have shown themselves in the accelerated 
 motion of the planets. The entire loss of heat from 
 the sun cannot at all events be produced in this way ; 
 at the most a portion, which, however, may not be 
 inconsiderable. 
 
 If, now, there is no present manifestation of force 
 sufficient to cover the expenditure of the sun's heat, 
 the sun must originally have had a store of heat which 
 it gradually gives out. But whence this store ? We 
 know that the cosmical forces ^lone could have pro- 
 duced it. And here the hypothesis, previously dis- 
 cussed as to the origin of the sun, comes to our aid. If 
 the mass of the sun had been once diffused in cosmical 
 space, and had then been condensed that is, had fallen 
 together under the influence of celestial gravity if 
 then the resultant motion had been destroyed by 
 friction and impact, with the production of heat, the 
 new world produced by such condensation must have 
 acquired a store of heat not only of considerable, but 
 even of colossal, magnitude. 
 
 Calculation shows that, assuming the thermal capa- 
 city of the sun to be the same as that of water, the 
 
182 ON THE ORIGIN OF THE PLANETAKY SYSTEM. 
 
 temperature might be raised to 28,000,000 of degrees, 
 if this quantity of heat could ever have been present 
 in the sun at one time. This cannot be assumed, for 
 such an increase of temperature would offer the 
 greatest hindrance to condensation. It is probable 
 rather that a great part of this heat, which was pro- 
 duced by condensation, began to radiate into space 
 before this condensation was complete. But the heat 
 which the sun could have previously developed by its 
 condensation, would have been sufficient to cover its 
 present expenditure for not less than 22,000,000 of 
 years of the past. 
 
 And the sun is by no means so dense as it may 
 become. Spectrum analysis demonstrates the presence 
 of large masses of iron and of other known constituents 
 of the rocks. The pressure which endeavours to con- 
 dense the interior is about 800 times as great as that 
 in the centre of the earth ; and yet the density of the 
 sun, owing probably to its enormous temperature, is 
 less than a quarter of the mean density of the earth. 
 
 We may therefore assume with great probability 
 that the sun will still continue in its condensation, even 
 if it only attained the density of the earth though it 
 will probably become far denser in the interior owing 
 to the enormous pressure this would develop fresh 
 quantities of heat, which would be sufficient to main- 
 tain for an additional 17,000,000 of years the same 
 
ON THE ORIGIN OF THE PLANETARY SYSTEM. 183 
 
 intensity of sunshine as that which is now the source 
 of all terrestrial life. 
 
 The smaller bodies of our system might become 
 less hot than the sun, because the attraction of the 
 fresh masses would be feebler. A body like the earth 
 might, if even we put its thermal capacity as high as 
 that of water, become heated to even 9,000 degrees > 
 to more than our flames can produce. The smaller 
 bodies must cool more rapidly as long as they are still 
 liquid. The increase in temperature, with the depth, 
 is shown in bore-holes and in mines. The existence of 
 hot wells and of volcanic eruptions shows that in the 
 interior of the earth there is a very high temperature,, 
 which can scarcely be anything than a remnant of the 
 high temperature which prevailed at the time of its 
 production. At any rate, the attempts to discover for 
 the internal heat of the earth a more recent origin in 
 chemical processes, have hitherto rested on very arbi- 
 trary assumptions ; and, compared with the general uni- 
 form distribution of the internal heat, are somewhat 
 insufficient. 
 
 On the other hand, considering the huge masses of 
 Jupiter, of Saturn, of Uranus, and of Neptune, their 
 small density, as well as that of the sun, is surprising, 
 while the smaller planets and the moon approximate to 
 the density of the earth. We are here reminded of 
 the higher initial temperature, and the slower cooling, 
 
184 ON THE ORIGIN OF THE PLANETARY SYSTEM. 
 
 which characterises larger masses. 1 The moon, on the 
 contrary, exhibits formations on its surface which are 
 strikingly suggestive of volcanic craters, and point to 
 a former state of ignition of our satellite. The mode 
 of its rotation, moreover, that it always turns the 
 same side towards the earth, is a peculiarity which 
 might have been produced by the friction of a fluid. 
 At present no trace of such a one can be perceived. 
 You see, thus, by what various paths we are con- 
 
 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 its 
 own. 
 
ON THE ORIGIN OF THE PLANETARY SYSTEM. 185 
 
 that all stages of that process can still be found to 
 exist. 
 
 For as we have already seen, the larger bodies 
 which are already formed go on increasing with the 
 
 FIG. 13. 
 
 development of heat, by the attraction of the meteoric 
 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 
 
186 ON THE ORIGIN OF THE PLANETARY SYSTEM. 
 
 those whose light is sufficiently strong give for the 
 most part a coloured spectrum of fine bright lines, a& 
 they appear in the spectra of the ignited gases. The 
 nebulae are partly rounded structures, which are called 
 
 FIG. 14. 
 
 planetary nebulce (fig. 12) ; sometimes wholly irregular 
 in form, as the large nebula in Orion, represented in 
 fig. 13; they are partly annular, as in the figures in 
 fig. 14, from the Canes Venatici. They are for the 
 most part feebly luminous over their whole surface, 
 while the fixed stars only appear as luminous points. 
 
ON THE ORIGIN OF THE PLANETARY SYSTEM. 187 
 
 -* 
 
 In many nebulae small stars can be seen, as in figs. 
 15 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 only heaps of 
 
 PIG. 15, FIG. 16. 
 
 very fine stars, of other Milky Ways. Now, however, 
 spectrum analysis has shown a gas spectrum in many 
 nebulae which contains stars, while actual heaps of stars 
 show the continuous spectrum of ignited solid bodies. 
 Nebulae have in general three distinctly recognisable 
 lines, one of which, in the blue, belongs to hydrogen, 
 a second in bluish-green to nitrogen, 1 while the third, 
 between the two, is of unknown origin. Fig. 1 7 shows 
 
 1 Or perhaps also to oxygen. The line occurs in the spectrum of 
 atmospheric air, and according to H. C. Vogel's observations was want- 
 ing in the spectrum of pure oxygen. 
 
188 ON THE OKIGIN OF THE PLANETARY SYSTEM. 
 
 such a spectrum of a small but bright nebula in the 
 Dragon. Traces of other bright lines are seen along 
 with them, and sometimes also, as in fig. 17, traces 
 of a continuous spectrum ; all of which, however, are 
 too feeble to admit of accurate investigation. It must 
 be observed here that the light of very feeble objects 
 which give a continuous spectrum are distributed by 
 the spectroscope over a large surface, and are there- 
 fore greatly enfeebled or even extinguished, while the 
 
 FIG. 17. 
 
 undecomposable light of bright gas lines remains unde- 
 composed, and hence can still be seen. In any case, 
 the decomposition of the light of the nebulae shows 
 that by far the greater part of their luminous surface 
 is due to ignited gases, of which hydrogen forms a 
 prominent constituent. In the planetary masses, the 
 spherical or discoidal, it might be supposed that the 
 gaseous mass had attained a condition of equilibrium ; 
 but most other nebulas exhibit highly irregular forms, 
 which by no means correspond to such a condition. 
 As, however, their shape has either not at all altered, 
 or not appreciably, since they have been known and 
 observed, they must either have very little mass, or 
 
ON THE ORIGIN OF THE PLANETARY SYSTEM. 189 
 
 they must be of colossal size and distance. The former 
 does not appear very probable, because small masses 
 very soon give out their heat, and hence we are left 
 to the second alternative, that they are of huge di- 
 mensions and distances. The same conclusion had 
 been originally drawn by Sir W. Herschel, on the 
 assumption that the nebulas were heaps of stars. 
 
 With those nebulae which, besides the lines of gases, 
 also show the continuous spectrum of ignited denser 
 bodies, are connected spots which are partly irresolv- 
 able and partly resolvable into heaps of stars, which 
 only show the light of the latter kind. 
 
 The countless luminous stars of the heavenly firma- 
 ment, whose number increases with each newer and 
 more perfect telescope, associate themselves with this 
 primitive condition of the worlds as they are formed. 
 They are like our sun in magnitude, in luminosity, and 
 on the whole also in the chemical condition of their 
 surface, although there may be differences in the quan- 
 tity of individual elements. 
 
 But we find also in space a third stadium, that of 
 extinct suns ; and for this also there are actual evi- 
 dences. In the first place, there are, in the course of 
 history, pretty frequent examples of the appearance of 
 new stars. In 1572 Tycho Brahe observed such a one, 
 which, though gradually burning paler, was visible 
 for two years, stood still like a fixed star, and finally 
 
190 ON THE ORIGIN OF THE PLANETARY SYSTEM. 
 
 reverted to the darkness from which it had so suddenly 
 emerged. The largest of them all seems to have been 
 that observed by Kepler in the year 1604, which was 
 brighter than a star of the first magnitude, and was 
 observed from September 27, 1604, until March 1606. 
 The reason of its luminosity was probably the collision 
 with a smaller world. In a more recent case, in which 
 on May 12, 1866, a small star of the tenth magnitude 
 in the Corona suddenly burst out to one of the second 
 magnitude, spectrum analysis showed that it was an 
 outburst of ignited hydrogen which produced the light. 
 This was only luminous for twelve days. 
 
 In other cases obscure heavenly bodies have dis- 
 covered themselves by their attraction on adjacent 
 bright stars, and the motions of the latter thereby pro- 
 duced. Such an influence is observed in Sirius and 
 Procyon. By means of a new refracting telescope 
 Messrs. Alvan Clarke and Pond, of Cambridge, U.S., 
 have discovered in the case of Sirius a scarcely visible 
 star, which has but little luminosity, but is almost 
 seven times as heavy as the sun, has about half the 
 mass of Sirius, and whose distance from Sirius is about 
 equal to that of Neptune from the sun. The satellite 
 of Procyon has not yet been seen; it appears to be 
 quite dark. 
 
 Thus there are extinct suns. The fact that there 
 are such lends new weight to the reasons which per- 
 
ON THE ORIGIN OF THE PLANETARY SYSTEM. 191 
 
 mit us to conclude that our sun also is a body which 
 slowly gives out its store of heat, and thus will some 
 time become extinct. 
 
 The term of 17,000,000 years which I have given 
 may perhaps become considerably prolonged by the 
 gradual abatement of radiation, by the new accretion of 
 falling meteors, and by still greater condensation than 
 that which I have assumed in that calculation. But 
 we know of no natural process which could spare our 
 sun the fate which has manifestly fallen upon other 
 suns. This is a thought which we only reluctantly 
 admit; it seems to us an insult to the beneficent 
 Creative Power which we otherwise find at work in 
 organisms and especially in living ones. But we must 
 reconcile ourselves to the thought that, however we 
 may consider ourselves to be the centre and final 
 object of Creation, we ate 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 in 
 
192 ON THE ORIGIN OF THE PLANETAEY SYSTEM. 
 
 process of formation, and will still continue to form 
 when our sun is extinguished, and our earth is either 
 solidified in cold or is united with the ignited central 
 body of our system. 
 
 But who knows whether the first living inhabitants 
 of the warm sea on the young world, whom we ought 
 perhaps to honour as our ancestors, would not have 
 regarded our present cooler condition with as much 
 horror as we look on a world without a sun ? Consider- 
 ing the wonderful adaptability to the conditions of life 
 which all organisms possess, who knows to what degree 
 of perfection our posterity will have been developed in 
 17,000,000 of years, and whether our fossilised bones 
 will not perhaps seem to them as monstrous as those of 
 the Ichthyosaurus now do ; and whether they, adjusted 
 for a more sensitive state of equilibrum, will not con- 
 sider the extremes of temperature, within which we now 
 exist, to be just as violent and destructive as those of the 
 older geological times appear to us ? Yea, even if sun and 
 earth should solidify and become motionless, who could 
 say what new worlds would not be ready to develop 
 life ? Meteoric stones sometimes contain hydrocarbons ; 
 the light of the heads of comets exhibits a spectrum 
 which is most like that of the electrical light in gases 
 containing hydrogen and carbon. But carbon is the 
 element, which is characteristic of organic compounds, 
 from which living bodies are built up. Who knows 
 
ON THE ORIGIN OF THE PLANETARY SYSTEM. 193 
 
 whether these bodies, which everywhere swarm through 
 space, do not scatter germs of life wherever there is a 
 new world, which has become capable of giving a dwell- 
 ing-place to organic bodies ? And this life we might 
 perhaps consider as allied to ours in its primitive 
 germ, however different might be the form which it 
 would assume in adapting itself to its new dwelling- 
 place. 
 
 However this may be, that which most arouses our 
 moral feelings at the thought of a future, though pos- 
 sibly very remote, cessation of all living creation on the 
 earth, is more particularly the question whether all this 
 life is not an aimless sport, which will ultimately fall a 
 prey to destruction by brute force ? Under the light of 
 Darwin's great thought we begin to see that not only 
 pleasure and joy, but also pain, struggle, and death, are 
 the powerful means by which nature has built up her 
 finer and more perfect forms of life. And we men 
 know more particularly that in our intelligence, our 
 civic order, and our morality we are living on the in- 
 heritance which our forefathers 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 
 thread of his own consciousness will one day break.. 
 
 II. 
 
194 ON THE ORIGIN OF THE PLANETARY SYSTEM. 
 
 But even men of such free and large order of minds 
 as Lessing and David Strauss could not reconcile them- 
 selves to the thought of a final destruction of the 
 living race, and with it of all the fruits of all past 
 generations. 
 
 As yet we know of no fact, which can be established 
 by scientific observation, which would show that the 
 finer and complex forms of vital motion could exist 
 otherwise than in the dense material of organic life ; 
 that it can propagate itself as the sound-movement 
 of a string can leave its originally narrow and fixed 
 home and diffuse itself in the air, keeping all the time 
 its pitch, and the most delicate shade of its colour-tint ; 
 and that, when it meets another string attuned to it, 
 starts this again or excites a flame ready to sing to the 
 same tone. The flame even, which, of all processes in 
 inanimate nature, is the closest type of life, may 
 become extinct, but the heat which it produces con- 
 tinues to exist indestructible, imperishable, as an in- 
 visible motion, now agitating the molecules of ponder- 
 able matter, and then radiating into boundless space as 
 the vibration of an ether. Even there it retains the 
 characteristic peculiarities of its origin, and it reveals its 
 history to the inquirer who questions it by the spectro- 
 scope. United afresh, these rays may ignite a new 
 flame, and thus, as it were, acquire a new bodily 
 existence. 
 
ON THE ORIGIN OF THE PLANETARY SYSTEM. 195 
 
 Just as the flame remains the same in appearance 
 and continues to exist with the same form and struc- 
 ture, although it draws every minute fresh combustible 
 vapour, and fresh oxygen from the air, into the vortex 
 of its ascending current ; and just as the wave goes 
 on in unaltered form, and is yet being reconstructed 
 every moment from fresh particles of water, so also in 
 the living being, it is not the definite mass of substance, 
 which now constitutes the body, to which the con- 
 tinuance of the individual is attached. For the material 
 of the body, like that of the flame, is subject to con- 
 tinuous and comparatively rapid change a change the 
 more rapid, the livelier the activity of the organs in 
 question. Some constituents are renewed from day to 
 day, some from month to month, and others only after 
 years. That which continues to exist as a particular 
 individual is like the flame and the wave only the 
 form of motion which continually attracts fresh matter 
 into its vortex and expels the old. The observer with 
 a deaf ear only recognises the vibration of sound 
 as long as it is visible and can be felt, bound up with 
 heavy matter. Are our senses, in reference to life, like 
 the deaf ear in this respect ? 
 
 o 2 
 
196 ON THE ORIGIN OF THE PLANETARY SYSTEM. 
 
 ADDENDUM. 
 
 THE sentences on page 193 gave rise to a controversial 
 attack by Mr. J. C. F. Zoellner, in his book * On the 
 Nature of the Comets,' on Sir W. Thomson, on which I 
 took occasion to express myself briefly in the preface to the 
 second part of the German translation of the ' Handbook of 
 Theoretical Physics/ by Thomson and Tait. I give here the 
 passage in question : 
 
 ' I will mention here a farther objection. It refers to the 
 question as to the possibility that organic germs may occur 
 in meteoric stones, and be conveyed to the celestial bodies 
 which have been cooled. In his opening Address at the 
 Meeting of the British Association in Edinburgh, in August 
 1871, Sir W. Thomson had described this as " not unscien- 
 tific." Here also, if there is an error, I must confess that I 
 also am a culprit. I had mentioned the same view as a 
 possible mode of explaining the transmission of organisms 
 through space, even a little before Sir "W. Thomson, in a 
 lecture delivered in the spring of the same year at Heidel- 
 berg and Cologne, but not published. I cannot object if any- 
 one considers this hypothesis to be in a high, or even in the 
 highest, degree improbable. But to me it seems a perfectly 
 correct scientific procedure, that when all our attempts fail 
 in producing organisms from inanimate matter, we may 
 inquire whether life has ever originated at all or not, and 
 whether its germs have not been transported from one 
 world to another, and have developed themselves wherever 
 they found a favourable soil. 
 
 1 Mr. Zoellner's so-called physical objections are but of 
 very small weight. He recalls the history of meteoric stone, 
 and adds (p. xxvi.): " If, therefore, that meteoric stones covered 
 with organisms had escaped with a whole skin in the smash- 
 
ON THE ORIGIN OF THE PLANETARY SYSTEM. 197 
 
 up of its mother-body, and had not shared the general rise 
 of temperature, it must necessarily have first passed through 
 the atmosphere of the earth, before it could deliver itself of 
 its organisms for the purpose of peopling the earth." 
 
 ' Now, in the first place, we know from repeated observa- 
 tions that the larger meteoric stones only become heated in 
 their outside layer during their fall through the atmosphere, 
 while the interior is cold, or even very cold. Hence all 
 germs which there might be in the crevices would be safe 
 from combustion in the earth's atmosphere. But even those 
 germs which were collected on the surface when they reached 
 the highest and most attenuated layer of the atmosphere would 
 long before have been blown away by the powerful draught 
 of air, before the stone reached the denser parts of the gaseous 
 mass, where the compression would be sufficient to produce 
 an appreciable heat. And, on the other hand, as far as the 
 impact of two bodies is concerned, as Thomson assumes, 
 the first consequences would be powerful mechanical motions, 
 and only in the degree in which this would be destroyed by 
 friction would heat be produced. We do not know whether 
 that would last for hours, for days, or for weeks. 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.' 
 
L I B H A ii Y 
 
 UNIVERSITY OK 
 
 CAL1FOKNIA. 
 
 ON 
 
 THOUGHT IN MEDICINE. 
 
 An Address delivered August 2, 1877, on the Anniversary of 
 
 the Foundation of the Institute for the Education of 
 
 Army Surgeons. 
 
 IT is now thirty-five years since, on the 2nd August, I 
 stood on the rostrum in the Hall of this Institute, before 
 another such audience as this, and read a paper on 
 the operation of Venal Tumours. I was then a pupil of 
 this Institution, and was just at the end of my studies. 
 I had never seen a tumour cut, and the subject-matter 
 of my lecture was merely compiled from books ; but 
 book knowledge played at that time a far wider and 
 a far more influential part in medicine than we are at 
 present disposed to assign to it. It was a period of 
 fermentation, of the fight between learned tradition and 
 the new spirit of natural science, which would have 
 no more of tradition, but wished to depend upon 
 individual experience. The authorities at that time 
 
200 ON THOUGHT IN MEDICINE. 
 
 judged more favourably of my Essay than I did myself, 
 and I still possess the books which were awarded to me 
 as the prize. 
 
 The recollections which crowd in upon me on this 
 occasion have brought vividly before my mind a picture 
 of the then condition of our science, of our endeavours 
 and of our hopes, and have led me to compare the 
 past state of things with that into which it has de- 
 veloped. Much indeed has been accomplished. 
 
 Although all that we hoped for has not been ful- 
 filled, and many things have turned out differently from 
 what we wished, yet we have gained much for which we 
 could not have dared to hope. Just as the history 
 of the world has made one of its few giant steps 
 before our eyes, so also has our science ; hence an old 
 student, like myself, scarcely recognises the somewhat 
 ir.atronly 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 
 slow steps. This must be my excuse for speaking to 
 
/^v '/<> 
 
 ON THOUGHT IN MEDICINE. /'/2 01 <l 
 
 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 cf thought of a period which 
 is so far behind us. The course of development of 
 medicine is an instructive lesson on the true principles 
 of scientific inquiry, and the positive part of this 
 lesson has, perhaps, in no previous time been so im- 
 pressively taught as in the last generation. 
 
 The task falls to me, of teaching that branch of the 
 natural sciences which has to make the widest gene- -- 
 ralisations, and has to discuss the meaning of funda- 
 mental ideas ; and which has, on that account, been 
 not unfitly termed Natural Philosophy by the English- 
 speaking peoples. Hence it does not fall too far out of 
 the range of my official duties and of my own studies, if I 
 attempt to discourse here of the principles of scientific 
 method, in reference to the sciences of experience. 
 
 As regards my acquaintance with the tone of 
 thought of the older medicine, independently of the 
 general obligation, incumbent on every educated 
 
202 ON THOUGHT IN MEDICINE. 
 
 physician, of understanding the literature of his science 
 and the direction as well as the conditions of its 
 progress, there was in my case a special incentive. In 
 my first professorship at Konigsberg, from the year 
 1849 to 1856, I had to lecture each winter on general 
 pathology that is, on that part of the subject which 
 contains the general theoretical conceptions of /'the 
 nature of disease, and of the principles of its treatment. 
 
 General pathology was regarded by our elders as 
 the fairest blossom of medical science. But in fact, 
 that which formed its essence possesses only historical 
 interest for the disciples of modern natural science. 
 
 Many of my predecessors have broken a lance for 
 the scientific defence of this essence, and more especially 
 Henle and Lotz. The latter, whose starting-point was 
 also medicine, had, in his general pathology and thera- 
 peutics, arranged it very thoroughly and methodically 
 and with great critical acumen. 
 
 My own original inclination was towards physics ; 
 external circumstances compelled me to commence the 
 study of medicine, which was made possible to me by 
 the liberal arrangements of this Institution. It had, 
 however, been the custom of a former time to combine 
 the study of medicine with that of the Natural Sciences, 
 and whatever in this was compulsory I must consider 
 fortunate ; not merely that I entered medicine at a 
 time in which any one who was even moderately at 
 
ON THOUGHT IN MEDICINE. 203 
 
 home 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 and the fearful practical scope of the problems 
 of medical theory, who has watched the fading eye of 
 approaching death, and witnessed the distracted grief 
 of affection, and who has asked himself the solemn 
 questions, Has all been done which could be done to 
 ward off the dread event ? Have all the resources and 
 all the means which Science has accumulated become 
 exhausted ? 
 
 Provided that he remains undisturbed in his study, 
 the purely theoretical inquirer may smile with calm 
 contempt when, for a time, vanity and conceit seek 
 to swell themselves in science and stir up a commo- 
 tion. Or he may consider ancient prejudices to be 
 interesting and pardonable, as remains of poetic ro- 
 mance, or of youthful enthusiasm. To one who has to 
 contend with the hostile forces of fact, indifference 
 and romance disappear ; that which he knows and can 
 do, is exposed to severe tests ; he can only use the 
 
204 ON THOUGHT IN MEDICINE. 
 
 hard and clear light of facts, and must give up the notion 
 of lulling himself in agreeable illusions. 
 
 I rejoice, therefore, that I can once more address 
 an assembly consisting almost exclusively of medical 
 men who have gone through the same school. Medicine 
 was once the intellectual home in which I grew up, and 
 even the emigrant best understands and is best under- 
 stood by his native land. 
 
 If I am called upon to designate in one word the 
 fundamental error of that former time, I should be in- 
 clined to say that it pursued a false ideal of science in 
 a one-sided and erroneous reverence for the deductive 
 method. Medicine, it is true, was not the only science 
 which was involved in this error, but in no other 
 science have the consequences been so glaring, or have 
 so hindered progress, as in medicine. The history of 
 this science claims, therefore, a special interest in the 
 history of the development of the human mind. None 
 other is, perhaps, more fitted to show that a true 
 criticism of the sources of cognition is also prac- 
 tically an exceedingly important object of true philo- 
 sophy. 
 
 The proud word of Hippokrates, 
 
 irjrpot; <f)t\6(TO(f)O iaoQeoQ, 
 
 6 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 we 
 
ON THOUGHT IN" MEDICINE. 205 
 
 are to understand as a philosopher. For the ancients, 
 philosophy embraced all theoretical knowledge ; their 
 philosophers pursued Mathematics, Physics, Astronomy, 
 Natural History, in close connection with true philo- 
 sophical or metaphysical considerations. If, therefore, 
 we are to understand the medical philosopher of Hip- 
 pokrates to be a man who has a perfected insight into 
 the causal connection of natural processes, we shall in 
 fact be able to say with Hippokrates, Such a one can 
 give help like a god. 
 
 Understood in this sense, the aphorism describes 
 in three words the ideal which our science has to strive 
 after. But who can allege that it will ever attain 
 this ideal ? 
 
 But those disciples of medicine who thought them- 
 selves divine even in their own lifetime, and who 
 wished to impose themselves upon others as such, were 
 not inclined to postpone their hopes for so long a 
 period. The requirements for the QiXocrofos were 
 considerably moderated. Every adherent of any given 
 cosmological system, in which, for well or ill, facts 
 must be made to correspond with reality, felt himself to 
 be a philosopher. The philosophers of that time knew 
 little more of the laws of Nature than the unlearned 
 layman ; but the stress of their endeavours was laid upon 
 thinking, upon the logical consequence and complete- 
 ness of the system. It is not difficult to understand 
 
206 ON THOUGHT IN MEDICINE. 
 
 how in periods of youthful development, such a one- 
 sided over-estimate of thought could be arrived at. 
 The superiority of man over animals, of the scholar 
 over the barbarian, depends upon thinking ; sensation, 
 feeling, perception, on the contrary, he shares with his 
 lower fellow-creatures, and in acuteness of the senses 
 many of these are even superior to him. That man 
 strives to develop his thinking faculty to the utmost 
 is a problem on the solution of which the feeling of 
 his own dignity, as well as of his own practical power, 
 depends ; and it is a natural error to have considered 
 unimportant the dowry of mental capacities which 
 Nature had given to animals, and to have believed that 
 thought could be liberated from its natural basis, 
 observation and perception, to begin its Icarian flight 
 of metaphysical speculation. 
 
 It is, in fact, no easy problem to ascertain com- 
 pletely the origins of our knowledge. An enormous 
 amount is transmitted by speech and writing. This 
 power which man possesses of gathering together the 
 stores of knowledge of generations, is the chief reason 
 of his superiority over the animal, who is restricted 
 to an inherited blind instinct and to its individual 
 experience. But all transmitted knowledge is handed 
 on already formed; whence the reporter has derived 
 it, or how much criticism he has bestowed upon it, 
 can seldom be made out, especially if the tradition has 
 
OX THOUGHT IN MEDICINE. 207 
 
 been handed down through several generations. We 
 must admit it all upon good faith; we cannot arrive 
 at the source ; and when many generations have con- 
 tented themselves with such knowledge, have brought 
 no criticism to bear upon it ; have, indeed, gradually 
 added all kinds of small alterations, which ultimately 
 grew up to large ones after all this, strange things are 
 often reported and believed under the authority of 
 primeval wisdom. A curious case of this kind is the 
 history of the circulation of the blood, of which we 
 shall still have to speak. 
 
 But another kind of tradition by speech, which 
 long remained undetected, is even still more confusing 
 for one who reflects upon the origin of knowledge. 
 Speech cannot readily develop names for classes of 
 objects or for classes of processes, if we have not been ac- 
 customed very often to mention together the correspond- 
 ing individuals, things, and separate cases, and to assert 
 what there is in common about them. They must, 
 therefore, possess many points in common. Or if we, 
 reflecting scientifically upon this, select some of these 
 characteristics, and collate them to form a definition, 
 the common possession of these selected characteristics 
 must necessitate that in the given cases a great num- 
 ber of other characteristics are to be regularly met 
 with ; there must be a natural connection between the 
 first and the last-named characteristics. If, for instance, 
 
208 ON THOUGHT IN MEDICINE. 
 
 we assign the name of mammals to those animals which, 
 when young, are suckled by their mothers, we can 
 assert further, in reference to them, that they are all 
 warm-blooded animals, born alive, that they have a 
 spinal column but no quadrate bone, breathe through 
 lungs, have separate divisions of the heart, &c. Hence 
 the fact, that in the speech of an intelligent observing 
 people a certain class of things are included in one 
 name, indicates that these things or cases fall under a 
 common natural relationship ; by this alone a host of 
 experiences are transmitted from preceding generations 
 without this appearing to be the case. 
 
 The adult, moreover, when he begins to reflect upon 
 the origin of his knowledge, is in possession of a huge 
 mass of every-day experiences, which in great part 
 reach back to the obscurity of his first childhood. 
 Everything individual has long been forgotten, but 
 the similar traces which the daily repetition of similar 
 cases has left in his memory have deeply engraved 
 themselves. And since only that which is in con- 
 formity with law is always repeated with regularity, 
 these deeply impressed remains of all previous con- 
 ceptions are just the conceptions of what is conform- 
 able to law in the things and processes. 
 
 Thus man, when he begins to reflect, finds that he 
 possesses a wide range of acquirements of which he 
 knows not whence they came, which he has possessed 
 
ON THOUGHT IN MEDICINE. 209 
 
 as long as he 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 
 
 II. P 
 
210 ON THOUGHT IN MEDICINE. 
 
 pupils, who copied their master, exaggerated his theory, 
 made it more one-sided and more logical, without 
 regard to any discordance with Nature. The more 
 rigid the system, the fewer and the more thorough 
 .were the methods to which the healing art was re- 
 stricted. The more the schools were driven into a 
 corner by the increase in actual knowledge, the more 
 did they depend upon the ancient authorities, and the 
 more intolerant were they against innovation. The 
 great reformer of anatomy, Vesalius, was cited before 
 the Theological faculty of Salamanca ; Servetus was 
 burned at Greneva along with his book, in which he 
 described the circulation of the lungs ; and the Paris 
 faculty prohibited the teaching of Harvey's doctrine of 
 the circulation of the blood in its lecture rooms. 
 
 At the same time the bases of the systems from 
 which these schools started were mostly views on 
 natural science which it would have been quite right 
 to utilise within a narrow circle. What was not 
 right was the delusion that it was more scientific to 
 refer all diseases to one kind of explanation, than to 
 several. What was called the solidar pathology wanted to 
 deduce everything from the altered mechanism of the 
 solid parts, especially from their altered tension ; from 
 the strictum and laxum, from tone and want of tone, 
 and afterwards from strained or relaxed nerves and from 
 obstructions in the vessels. Humoral pathology was 
 
ON THOUGHT IN MEDICINE. 211 
 
 only acquainted with alterations in mixture. The four 
 cardinal fluids, representatives of the classical four 
 elements, blood, phlegm, black and yellow gall ; with 
 others, the acrimonies or dyscrasies, which had to be 
 expelled by sweating and purging ; in the beginning of 
 our modern epoch, the acids and alkalies or the alchy- 
 mistic spirits, and the occult qualities of the substances 
 assimilated all these were the elements of this chem- 
 istry. Along with these were found all kinds of phy- 
 siological conceptions, some of which contained remark- 
 able foreshadowings, such as the sfi^vrov Qsp/juov, 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 irvevfta, 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- 
 
 p 2 
 
212 ON THOUGHT IN MEDICINE. 
 
 servation 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 
 
OX THOUGHT IN MEDICINE. 213 
 
 the principle from which it is deduced ; and that each 
 new induction must in the first place be a new test, by 
 experience, of its own bases. That a conclusion is de- 
 duced by the strictest logical method from an uncertain 
 premiss does not give it a hair's breadth of certainty 
 or of value. 
 
 One characteristic of the schools which built up 
 their system on such hypotheses, which they assumed 
 as dogmas, is the intolerance of expression which I have 
 already partially mentioned. One who works upon a well- 
 ascertained foundation may readily admit an error ; he 
 loses, by so doing, nothing more than that in which he 
 erred. If, however, the starting-point has been placed 
 upon a hypothesis, which either appears guaranteed by 
 authority, or is only chosen because it agrees with that 
 which it is wished to believe true, any crack may then 
 hopelessly destroy the whole fabric of conviction. The 
 convinced disciples must therefore claim for each 
 individual part of such a fabric the same degree of 
 infallibility ; for the anatomy of Hippokrates just as 
 much as for fever crises; every opponent must only 
 appear then as stupid or depraved, and the dispute will 
 thus, according to old precedent, be so much the more 
 passionate and personal, the more uncertain is the basis 
 which is defended. We have frequent opportunities 
 of confirming these general rules in the schools of 
 dogmatic deductive medicine. They turned their in- 
 
214 ON THOUGHT IN MEDICINE. 
 
 tolerance partly against each other, and partly against 
 the eclectics who found various explanations for various 
 forms of disease. This method, which in its essence is 
 completely justified, had, in the eyes of systematists, 
 the defect of being illogical. And yet the greatest 
 physicians and observers, Hippokrates at the head, 
 Aretaeus, Galen, Sydenham, and Boerhaave, had become 
 eclectics, or at any rate very lax systematists. 
 
 About the time when we seniors commenced the 
 study of medicine, it was still under the influence of 
 the important discoveries which Albrecht von Haller had 
 made on the excitability of nerves ; and which he had 
 placed in connection with the vitalistic theory of the 
 nature of life. Haller had observed the excitability in 
 the nerves and muscles of amputated members. The 
 most surprising thing to him was, that the most varied 
 external actions, mechanical, chemical, thermal, to 
 which electrical ones were subsequently added, had 
 always the same result; namely, that they produced 
 muscular contraction. They were only quantitatively 
 distinguished as regards their action on the organism, 
 that is, only by the strength of the excitation; he 
 designated them by the common name of stimulus ; 
 he called the altered condition of the nerve the exci- 
 tation, and its capacity of responding to a stimulus 
 the excitability, which was lost at death. This entire 
 condition of things, which physically speaking asserts 
 
ON THOUGHT IK MEDICINE. 215 
 
 no more than that the nerves, as concerns the changes 
 which take place in them after excitation, are in an 
 exceedingly unstable state of equilibrium ; this was 
 looked upon as the fundamental property of animal 
 life, and was unhesitatingly transferred to the other 
 organs and tissues of the body, for which there was no 
 similar justification. It was believed that none of 
 them were active of themselves, but must receive an 
 impulse by a stimulus from without; air and nourish- 
 ment were considered to be the normal stimuli. The 
 kind of activity seemed, on the contrary, to be con- 
 ditioned by the specific energy of the organ, under the 
 influence of the vital force. Increase or diminution 
 of the excitability was the category under which the 
 whole of the acute diseases were referred, and from 
 which indications were taken as to whether the treat- 
 ment should be lowering or stimulating. The rigid 
 one-sidedness and the unrelenting logic with which 
 Robert Brown had once worked out this system was 
 broken, but it always furnished the leading points of 
 view. 
 
 The vital force had formerly lodged as ethereal 
 spirit, as a Pneuma in the arteries ; it had then with 
 Paracelsus acquired the form of an Archeus, a kind 
 of useful Kobold, or indwelling alchymist, and had 
 acquired its clearest scientific position as ' soul of 
 life,' anima inscia, in Georg Ernst Stahl, who, in 
 
216 ON THOUGHT IN MEDICINE. 
 
 the first half of 3 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 4 soul of life ' 
 is, on the whole, constructed on the pattern on which 
 the pietistic communities of that period represented to 
 themselves the sinful human soul; it is subject to 
 errors and passions, to sloth, fear, impatience, sorrow, 
 indiscretion, despair. The physician must first appease 
 it, or then incite it, or punish it, and compel it to 
 repent. And the way in which, at the same time, he 
 established the necessity of the physical and vital 
 actions was well thought out. The soul of life governs 
 the body, and only acts by means of the physico- 
 chemical forces of the substances assimilated. But it 
 has the power to bind and to loose these forces, to allow 
 them full play or to restrain them. After death the 
 restrained forces become free, and evoke putrefaction 
 and decomposition. For the refutation of this hypo- 
 
ON THOUGHT IN MEDICINE. 217 
 
 thesis of binding and loosing, it was necessary to dis- 
 cover the law of the conservation of force. 
 
 The second half of the previous century was too 
 much possessed by the principles of rationalism to recog- 
 nise openly Stahl's ' soul of life.' It was presented 
 more scientifically as vital force, Vis mtalis, 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 8ymptomata 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, 
 
218 ON THOUGHT IN MEDICINE. 
 
 which since that time has almost been entirely aban- 
 doned. Therapeutics became still more impoverished as 
 the younger and more critical generation grew up, and 
 tested the assumptions of that which was considered to 
 be scientific. Among the younger generation were 
 many who, in despair as to their science, had almost 
 entirely given up therapeutics, or on principle had 
 grasped at an empiricism such as Rademacher then 
 taught, which regarded any expectation of a scientific 
 explanation as a vain hope. 
 
 What we learned at that time were only the ruins 
 of the older dogmatism, but their doubtful features 
 soon manifested themselves. 
 
 The vitalistic physician considered that the essen- 
 tial part of the vital processes did not depend upon 
 natural forces, which, doing their work with blind 
 necessity and according to a fixed law, determined the 
 result. What these forces could do appeared quite 
 subordinate, and scarcely worthy of a minute study. 
 He thought that he had to deal with a soul-like being, 
 to which a thinker, a philosopher, and an intelligent 
 man must be opposed. May I elucidate this by a few 
 outlines ? 
 
 At this time auscultation and percussion of the 
 organs of the chest were being regularly practised in 
 the clinical wards. But I have often heard it main- 
 tained that they were a coarse mechanical means of 
 
ON THOUGHT IN MEDICINE. 219 
 
 investigation which a physician with a clear mental 
 vision did not need ; and it indeed lowered and debased 
 the patient, who was anyhow a human being, by treat- 
 ing him as a machine. To feel the pulse seemed the 
 most direct method of learning the mode of action of 
 the vital force, and it was practised, therefore, as by far 
 the most important means of investigation. To count 
 with a repeater was quite usual, but seemed to the 
 old gentlemen as a method not quite in good taste. 
 There was, as yet, no idea of measuring temperature in 
 cases of disease. In reference to the ophthalmoscope, a 
 celebrated surgical colleague said to me that he would 
 never use the instrument, it was too dangerous to admit 
 crude light into diseased eyes ; another said the mirror 
 might be useful for physicians with bad eyes, his, how- 
 ever, were good, and he did not need it. 
 
 A professor of physiology of that time, celebrated 
 for his literary activity, and noted as an orator and 
 intelligent man, had a dispute on the images in the eye 
 with his colleague the physicist. The latter challenged 
 the physiologist to visit him and witness the experi- 
 ment. The physiologist, however, refused his request 
 with indignation ; alleging that a physiologist had 
 nothing to do with experiments ; they were of no good 
 but for the physicist. Another aged and learned pro- 
 fessor of therapeutics, who occupied himself much with 
 the reorganisation of the Universities, was urgent with 
 
220 ON THOUGHT IN MEDICINE. 
 
 me to divide physiology, in order to restore the good 
 old time ; that I myself should lecture on the really 
 intellectual part, and should hand over the lower 
 experimental part to a colleague whom he regarded as 
 good enough for the purpose. He quite gave me up 
 when 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 
 not been imitated elsewhere. Yet medicine possessed 
 
ON THOUGHT IN MEDICINE. 221 
 
 in anatomical dissections a great means of education 
 for independent observation, which is wanting in the 
 other faculties, and to which I am disposed to attach 
 great weight. Microscopic demonstrations were iso- 
 lated and infrequent in the lectures. Microscopic 
 instruments were costly and scarce. I came into pos- 
 session of one by having spent my autumn vacation 
 in 1841 in the Charite, prostrated by typhoid fever; as 
 pupil, I was nursed without expense, and on my re- 
 covery I found myself in possession of the savings of 
 my small resources. The instrument was not beautiful, 
 yet I was able to recognise by its means the prolonga- 
 tions of the ganglionic cells in the invertebrata, which 
 I described in my dissertation, and to investigate the 
 vibrions in my research on putrefaction and fermenta- 
 tion. 
 
 Any of my fellow-students who wished to make 
 experiments had to do so at the cost of his pocket- 
 money. One thing we learned thereby, which the 
 younger generation does not, perhaps, learn so well in 
 the laboratories that is, to consider in all directions 
 the ways and means of attaining the end, and to ex- 
 haust all possibilities, in the consideration, until a prac- 
 ticable path was found. We had, it is true, an almost 
 uncultivated field before us, in which almost every 
 stroke of the spade might produce remunerative results. 
 
 It was one man more especially who aroused our 
 
222 ON THOUGHT IN MEDICINE. 
 
 enthusiasm for work in the right direction that is, 
 Johannes Miiller, the physiologist. In his theoretical 
 views he favoured the vitalistic hypothesis, but in the 
 most essential points he was a natural philosopher, firm 
 and immovable; for him, all theories were but hy- 
 potheses, which had to be tested by facts, and about 
 which facts could alone decide. Even the views upon 
 those points which most easily crystallise into dogmas, 
 on the mode of activity of the vital force and the activity 
 of the conscious soul, he tried continually to define more 
 precisely, to prove or to refute by means of facts. 
 
 And, although the art of anatomical investigation was 
 most familiar to him, and he therefore recurred most 
 willingly to this, yet he worked himself into the chemical 
 and physical methods which were more foreign to him. 
 He furnished the proof that fibrine is dissolved in blood ; 
 he experimented on the propagation of sound in such 
 mechanisms as are found in the drum of the ear; 
 he treated the action of the eye as an optician. His 
 most important performance for the physiology of the 
 nervous system, as well as for the theory of cognition, 
 was the actual definite establishment of the doctrine of 
 the specific energies of the nerves. In reference to the 
 separation of the nerves of motor and sensible energy, 
 he showed how to make the experimental proof of 
 Bell's law of the roots of the spinal cord so as to 
 be free from errors; and in regard to the sensible 
 
ON THOUGHT IN MEDICINE. 223 
 
 energies he not only established the general law, but 
 carried out a great number of separate investigations, to 
 eliminate objections, and to refute false indications and 
 evasions. That which hitherto had been imagined from 
 the data of every-day experience, and which had been 
 sought to be expressed in a vague manner, in which the 
 true was mixed up with the false ; or which had just 
 been established for individual branches, such as by Dr. 
 Young for the theory of colours, or by Sir Charles Bell 
 for the motor nerves, that emerged from Miiller's hands 
 in a state of classical perfection a scientific achieve- 
 ment whose value I am inclined to consider as equal to 
 that of the discovery of the law of gravitation. 
 
 His scientific tendency, and more especially his ex- 
 ample, were continued in his pupils. We had been 
 preceded by Schwann, Henle, Reichert, Peters, Remak ; 
 I met as fellow-students E. Du Bois-Reymond, Virchow, 
 Briicke, Ludwig, Traube, J. Meyer, Lieberkiihn, Hall- 
 mann ; we were succeeded by A. von Grraefe, 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 
 tendencies among Miiller's contemporaries, among whom 
 
224 ON THOUGHT IN MEDICINE. 
 
 the three brothers Weber of Leipzig must first of all be 
 mentioned, who have built solid foundations in the 
 mechanism of the circulation, of the muscles, of the 
 joints, and of the ear. 
 
 The attack was made wherever a way could be 
 perceived of understanding one of the vital processes ; 
 it was assumed that they could be understood, and 
 success justified this assumption. A delicate and 
 copious technical apparatus has been developed in the 
 methods of microscopy, of physiological chemistry, and 
 of vivisection ; the latter greatly facilitated more par- 
 ticularly by the use of anaesthetic ether and of the para- 
 lysing curara, by which a number of deep problems 
 became open to attack, which to our generation seemed 
 hopeless. The thermometer, the ophthalmoscope, the 
 auricular speculum, the laryngoscope, nervous irritation 
 on the living body, opened out to the physician possibi- 
 lities of delicate and yet certain diagnosis where there 
 seemed to be absolute darkness. The continually in- 
 creasing number of proved parasitical organisms substi- 
 tute tangible objects for mystical entities, and teach 
 the surgeon to forestall the fearfully subtle diseases of 
 decomposition. 
 
 But do not think, gentlemen, that the struggle is at 
 an end. As long as there are people of such astound- 
 ing conceit as to imagine that they can effect, by a few 
 clever strokes, that which man can otherwise only hope 
 
ON THOUGHT IN MEDICINE. 225 
 
 to achieve by toilsome labour, hypotheses will be started 
 which, propounded as dogmas, at once promise to 
 solve all riddles. And as long as there are people who 
 believe implicitly in that which they wish to be true, 
 so long will the hypotheses of the former find credence* 
 Both classes will certainly not die out, and to the latter 
 the majority will always belong. 
 
 There are two characteristics more particularly 
 which metaphysical systems have always possessed. 
 In the first place man is always desirous of feeling 
 himself to be a being of a higher order, far beyond the 
 standard of the rest of nature ; this wish is satisfied by 
 the spiritualists. On the other hand, he would like 
 to believe that by his thought he was unrestrained 
 lord of the world, and of course by his thinking with 
 those conceptions, to the development of which he 
 has attained; this is attempted to be satisfied by the 
 materialists. 
 
 But one who, like the physician, has actively to face 
 natural forces which bring about weal or woe, is also 
 under the obligation of seeking for a knowledge of 
 the truth, and of the truth only ; without considering 
 whether, what he finds, is pleasant in one way or the 
 other. His aim is one which is firmly settled ; for him 
 the success of facts is alone finally decisive. He must 
 endeavour to ascertain beforehand, what will be the 
 result of his attack if he pursues this or that course. 
 
 II. Q 
 
226 ON THOUGHT IN MEDICINE. 
 
 In order to acquire this foreknowledge of what is 
 coming, but of what has not been settled by obser- 
 vations, no other method is possible than that of 
 endeavouring to arrive at the laws of facts by observa- 
 tions; and we can only learn them by induction, by the 
 careful selection, collation, and observation of those cases 
 which fall under the law. When we fancy that we have 
 arrived at a law, the business of deduction commences. 
 It is then our duty to develop the consequences of our 
 law as completely as may be, but in the first place only 
 to apply to them the test of experience, so far as they 
 can be tested, and then to decide by this test whether 
 the law holds, and to what extent. This is a test 
 which really never ceases. The true natural philo- 
 sopher reflects at each new phenomenon, whether the 
 best established laws of the best known forces may 
 not experience a change ; it can of course only be a 
 question of a change which does not contradict the whole 
 store of our previously collected experiences. It never 
 thus attains unconditional truth, but such a high degree 
 of probability that it is practically equal to certainty. 
 The metaphysicians may amuse themselves at this ; we 
 will take their mocking to heart when they are in a 
 position to do better, or even as well. The old words 
 of Socrates, the prime master of inductive definitions, 
 in reference to them are just as fresh as they were 
 2,000 years ago : c They imagined they knew what they 
 
ON THOUGHT IN MEDICINE. 227 
 
 did not know, and he at any rate had the advantage 
 of not pretending to know what he did not know.' 
 And again, he was surprised at its not being clear to 
 them that it is not possible for men to discover such 
 things ; since even those who most prided themselves 
 on the speeches made on the matter, did not agree 
 among themselves, but behaved to each other like 
 madmen (rots ^aivo^svoLs 6 pottos). 1 Socrates calls 
 them TOVS fjusyLcrrov 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, Memorabil. I. i. 11. 
 
 2 Arthur Schopenhauer, Von ikm, uber ihti von Frauenstadt und 
 Lindner. Berlin, 1863, p. 653. 
 
 Q 2 
 
228 ON THOUGHT IN MEDICINE. 
 
 Hence all those aspire after it who wish to pass as 
 the favoured children of genius. It seems, too, so 
 easy, so free from trouble, to get by sudden mental 
 flashes an unattainable advantage over our contem- 
 poraries. The true artist and the true inquirer knows 
 that great works can only be produced by hard work. 
 The proof that the ideas formed do not merely scrape 
 together superficial resemblances, but are produced by 
 a quick glance into the connection of the whole, can 
 only be acquired when these ideas are completely de- 
 veloped that is, for a newly discovered natural law, 
 only by its agreement with facts. This estimate must 
 by no means be regarded as depending on external 
 success, but the success is here closely connected with 
 the depth and completeness of the preliminary per- 
 ceptions. 
 
 To find superficial resemblances is easy ; it is 
 amusing in society, and witty thoughts soon procure for 
 their author the name of a clever man. Among the 
 great number of such ideas, there must be some which 
 are ultimately found to be partially or wholly correct ; 
 it would be a stroke of skill always to guess falsely. 
 In such a happy chance a man can loudly claim his 
 priority for the discovery ; if otherwise, a lucky 
 oblivion conceals the false conclusions. The adherents 
 of such a process are glad to certify the value of a first 
 thought. Conscientious workers who are shy at bring- 
 
ON THOUGHT IN MEDICINE. 229 
 
 ing their thoughts before the public before they have 
 tested them in all directions, solved all doubts, and 
 have firmly established the proof, these are at a decided 
 disadvantage. To settle the present kind of questions 
 of priority, only by the date of their first publication, 
 and without considering the ripeness of the research, 
 has seriously favoured this mischief. 
 
 In the * type case ' of the printer all the wisdom of 
 the world is contained which has been or can be dis- 
 covered ; it is only requisite to know how the letters 
 are to be arranged. So also, in the hundreds of books 
 and pamphlets which are every year published about 
 ether, the structure of atoms, the theory of perception, 
 as well as on the nature of the asthenic fever and 
 carcinoma, all the most refined shades of possible hy- 
 potheses are exhausted, and among these there must 
 necessarily be many fragments of the correct theory. 
 But who knows how to find them ? 
 
 I insist upon this in order to make clear to you that 
 all this literature, of untried and unconfirmed hypo- 
 theses, has no value in the progress of science. On the 
 contrary, the few sound ideas which they may contain 
 are concealed by the rubbish of the rest ; and one who 
 wants to publish something really new facts sees 
 himself open to the danger of countless claims of 
 priority, unless he is prepared to waste time and power 
 in reading beforehand a quantity of absolutely useless 
 
230 ON THOUGHT IN MEDICINE. 
 
 bookstand to de'stroy 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 Vernunft is a 
 continual sermon against the use of the category of 
 thought beyond the limits of possible experience. But 
 geometry seemed to him to do something which meta- 
 physics was striving after ; and hence geometrical 
 axioms, which he looked upon as a, priori principles 
 antecedent to all experience, he held to be given by 
 transcendental intuition, or as the inherent form of 
 all external intuition. Since that time, pure a priori 
 intuition has been the anchoring-ground of metaphy- 
 sicians. It is even more convenient than pure thought^ 
 because everything can be heaped on it without going 
 into chains of reasoning, which might be capable of 
 proof or of refutation. The nativistic theory of per- 
 ception of the senses is the expression of this theory 
 in physiology. All mathematicians united to fight 
 against any attempt to resolve the intuitions into their 
 
ON THOUGHT IN MEDICINE. 231 
 
 natural elements ; whether the so-called pure or the 
 empirical, the axioms of geometry, the principles of 
 mechanics, or the perceptions of vision. For this 
 reason, therefore, the mathematical investigations of 
 Lobatschewsky, Gauss, and Eiemann on the altera- 
 tions which are logically possible in the axioms of 
 geometry; and the proof that the axioms are principles 
 which are to be confirmed or perhaps even refuted by 
 experience, and can accordingly be acquired from ex- 
 perience these I consider to be very important steps. 
 That all metaphysical sects get into a rage about this 
 must not lead you astray, for these investigations lay 
 the axe at the bases of apparently the firmest supports 
 which their claims still possess. Against those investi- 
 gators who endeavour to eliminate from among the per- 
 ceptions of the senses, whatever there may be of the 
 actions of memory, and of the repetition of similar im- 
 pressions, which occur in memory ; whatever, in short, 
 is a matter of experience, against them it is attempted 
 to raise a party cry that they are spiritualists. As if 
 memory, experience, and custom were not also facts, 
 whose laws are to be sought, and which are not to be 
 explained away because they cannot be glibly referred 
 to reflex actions, and to the complex of the prolonga- 
 tion of ganglionic cells, and of the connection of nerve- 
 fibres in the brain. 
 
 Indeed, however self-evident, and however important 
 
232 ON THOUGHT IN MEDICINE. 
 
 the principle may appear to be, that natural science has 
 to seek for the laws of facts, this principle is neverthe- 
 less often forgotten. In recognising the law found, as a 
 force which rules the processes in nature, we conceive it 
 objectively as a, force, and such a reference of individual 
 cases to a force which under given conditions produces 
 a definite result, that we designate as a causal explana- 
 tion of phenomena. We cannot always refer to the 
 forces of atoms ; we speak of a refractive force, of electro- 
 motive and of electrodynamic force. But do not forget 
 the given conditions and the given result. If these 
 cannot be given, the explanation attempted is merely 
 a modest confession of ignorance, and then it is decidedly 
 better to confess this openly. 
 
 If any process in vegetation is referred to forces in 
 the cells, without a closer definition of the conditions 
 among which, and of the direction in which, they work, 
 this can at most assert that the more remote parts of 
 the organism are without influence ; but it would be 
 difficult to confirm this with certainty in more than a 
 few cases. In like manner, the originally definite sense 
 which Johannes Miiller gave to the idea of reflex action, 
 is 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 may be imposed upon the irresolvable 
 
ON THOUGHT IN MEDICINE. 233 
 
 complexity of the nerve-fibres of the brain. But the 
 resemblance to the qualitates occultce of ancient 
 medicine is very suspicious. 
 
 From the entire chain of my argument it fol- 
 lows that what I have said against metaphysics is 
 not intended against philosophy. But metaphysicians 
 have always tried to plume themselves on being philo- 
 sophers, and philosophical amateurs have mostly taken 
 an interest in the high-flying speculations of the meta- 
 physicians, by which they hope in a short time, and 
 at no great trouble, to learn the whole of what is worth 
 knowing. On another occasion * I compared the rela- 
 tionship of metaphysics to philosophy with that of 
 astrology to astronomy. The former had the most 
 exciting interest for the public at large, and especially 
 for the fashionable world, and turned its alleged con- 
 noisseurs into influential persons. Astronomy, on the 
 contrary, although it had become the ideal of scientific 
 research, had to be content with a small number of 
 quietly working disciples. 
 
 In like manner, philosophy, if it gives up meta- 
 physics, still possesses a wide and important field, the 
 knowledge of mental and spiritual processes and their 
 laws. Just as the anatomist, when he has reached the 
 limits of microscopic vision, must try to gain an in- 
 
 1 Preface to the German translation of Tyndall's Scientific 
 Fragments, p. xxii. 
 
234 ON THOUGHT IN MEDICINE. 
 
 sight into the action of his optical instrument, in like 
 manner every scientific enquirer must study minutely 
 the chief instrument of his research as to its capabili- 
 ties. The groping of the medical schools for the last 
 two thousand years is, among other things, an illus- 
 tration of the harm of erroneous views in this respect. 
 And the physician, the statesman, the jurist, the 
 clergyman, and the teacher, ought to be able to build 
 upon a knowledge of physical processes if they wish 
 to acquire a true scientific basis for their practical ac- 
 tivity. But the true science of philosophy has had, 
 perhaps, to suffer more from the evil mental habits and 
 the false ideals of metaphysics than even medicine 
 itself. 
 
 One word of warning. I should not like you to 
 think that my statements are influenced by personal 
 irritation. I need not explain that one who has such 
 opinions as I have laid before you, who impresses on 
 his pupils, whenever he can, the principle that ' a 
 metaphysical conclusion is either a false conclusion or 
 a concealed experimental conclusion,' that he is not 
 exactly beloved by the votaries of metaphysics or of 
 intuitive conceptions. Metaphysicians, like all those 
 who cannot give any decisive reasons to their oppo- 
 nents, are usually not very polite in their controversy ; 
 one's own success may approximately be estimated 
 from the increasing want of politeness in the replies. 
 
ON THOUGHT IN MEDICINE. 235 
 
 My own researches have led me more than other 
 disciples of the school of natural science into contro- 
 versial regions; and the expressions of metaphysical 
 discontent have perhaps concerned me even more than 
 my friends, as many of you are doubtless aware. 
 
 In order, therefore, to leave my own personal opinions 
 quite on one side, I have allowed two unsuspected war- 
 rantors to speak for me Socrates and Kant both of 
 whom were certain that all metaphysical systems estab- 
 lished up to their time were full of empty false con- 
 clusions, and who guarded themselves against adding 
 any new ones. In order to show that the matter has 
 not changed, either in the last 2,000 years or in the 
 last 100 years, let me conclude with a sentence of one 
 who was unfortunately too soon taken away from us, 
 Frederick Albert Lange, the author of the ' History of 
 Materialism.' In his posthumous 'Logical Studies,' 
 which he wrote in anticipation of his approaching end, 
 he gives the following picture, which struck me because 
 it would hold just as well in reference to solidar or 
 humoral pathologists, or any other of the old dogmatic 
 schools of medicine. 
 
 Lange says : The Hegelian ascribes to the Herbartian 
 a less perfect knowledge than to himself, and conversely ; 
 but neither hesitates to consider the knowledge of the 
 other to be higher compared with that of the empiricist, 
 and to recognise in it at any rate an approximation to 
 
236 ON THOUGHT IN MEDICINE. 
 
 the only true knowledge. It is seen, also, that here no 
 regard is paid to the validity of the proof, and that a 
 mere statement in the form of a deduction from the 
 entirety of a system is recognised as ' apodictic know- 
 ledge.' 
 
 Let us, then, throw no stones at our old medical 
 predecessors, who in dark ages, and with but slight 
 preliminary knowledge, fell into precisely the same 
 errors as the great intelligences of what wishes to be 
 thought the illuminated nineteenth century. They did 
 no worse than their predecessors except that the non- 
 sense of their method was more prominent in the 
 matter of natural science. Let us work on. In this work 
 of true intelligence physicians are called upon to play 
 a prominent part. Among those who are continually 
 called upon actively to preserve and apply their know- 
 ledge of nature, you are those who begin with the best 
 mental preparation, and are acquainted with the most 
 varied regions of natural phenomena. 
 
 In order, finally, to conclude our consultation on the 
 condition of Dame Medicine correctly with the epikri- 
 sis, I think we have every reason to be content with the 
 success of the treatment which the school of natural 
 science has applied, and we can only recommend the 
 younger generation to continue the same therapeutics. 
 
237 
 
 L I B R A R Y 
 
 -UNIVKKSITY OF 
 
 NX CALIFORNIA. 
 
 ON 
 
 ACADEMIC FEEEDOM 
 G-EEMAN UNIVEESITIES. 
 
 Inaugural Address as Rector of the Frederick William 
 University of Berlin. Delivered October 15, 1877. 
 
 IN entering upon the honourable office to which the 
 confidence of my colleagues has called me, my first 
 duty is once more openly to express my thanks to 
 those who have thus honoured me by their confidence. 
 I have the more reason to appreciate it highly, as it 
 was conferred upon me, notwithstanding that I have 
 been but few years among you, and notwithstanding 
 that I belong to a branch of natural science which 
 has come within the circle of University instruction 
 in some sense as a foreign element ; which has necessi- 
 tated many changes in the old order of University 
 teaching, and which will, perhaps, necessitate other 
 changes. It is indeed just in that branch (Physics) 
 
238 ON ACADEMIC FREEDOM IN GERMAN UNIVERSITIES. 
 
 which I represent, and which forms the theoretical 
 basis of all other branches of Natural Science, that 
 the particular characteristics of their methods are 
 most definitely pronounced. I have already been seve- 
 ral times in the position of having to propose altera- 
 tions in the previous regulations of the University, 
 and I have always had the pleasure of meeting with 
 the ready assistance of my colleagues in the faculty, 
 and of the Senate. That you have made me the 
 Director of the business of this University for this 
 year, is a proof that you regard me as no thought- 
 less innovator. For, in fact, however the objects, the 
 methods, the more immediate aims of investigations 
 in the natural sciences may differ externally from 
 those of the mental sciences, and however foreign 
 their results and however remote their interest may 
 often appear, to those who are accustomed only to the 
 direct manifestations and products of mental activity, 
 there is in reality, as I have endeavoured to show in my 
 discourse as Eector 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 out in the natural sciences far more promi- 
 
V f 
 
 ON ACADEMIC FJREEDOM IN GEKMAN UNIVEI^TIES./29 
 
 ^/ > ^ 
 
 nently that the real is far more sharply sep^ajbed 
 
 from the unreal, by the incorruptible criticism ety . 
 facts, than is the case with the more complex problems 
 of mental science. 
 
 And not merely the development of this new side 
 of scientific activity, which was almost unknown to 
 antiquity, but also the influence of many political, 
 social, and even international relationships make 
 themselves felt, and require to be taken into account. 
 The circle of our students has had to be increased; 
 a changed national life makes other demands upon 
 those who are leaving ; the sciences become more and 
 more specialised and divided ; exclusive of the libraries, 
 larger and more varied appliances for study are re- 
 quired. We can scarcely foresee what fresh demands 
 and what new problems we may have to meet in the 
 more immediate future. 
 
 On the other hand, the German Universities have 
 conquered a position of honour not confined to their 
 fatherland; the eyes of the civilised world are upon them. 
 Scholars speaking the most different languages crowd 
 towards them, even from the farthest parts of the 
 earth. Such a position would be easily lost by a false 
 step, but would be difficult to regain. 
 
 Under these circumstances it is our duty to get a 
 clear understanding of the reason for the previous pro- 
 sperity of our Universities ; we must try to find what is 
 
240 ON ACADEMIC FREEDOM IN GERMAN UNIVERSITIES. 
 
 the feature in their arrangements which we must seek 
 to retain as a precious jewel, and where, on the contrary, 
 we may give way when changes are required. I consider 
 myself by no means entitled to give a final opinion on 
 this matter. The point of view of any single indi- 
 vidual is restricted; representatives of other sciences 
 will be 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,' c Colleges,' 
 whose older members, the seniors, governed the com- 
 
ON ACADEMIC FREEDOM IN GERMAN UNIVERSITIES. 241 
 
 mon affairs of each such union, and also met together 
 for regulating the common affairs of the University. 
 In the courtyard of the University of Bologna are still 
 to be seen the eoats-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 
 unfrequently 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 
 
 II. R 
 
242 ON ACADEMIC FREEDOM IN GERMAN UNIVERSITIES. 
 
 that of Paifis, the most celebrated of all at the head 
 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 Koman and now of the Anglican 
 Church, whose instruction laymen might also share in 
 so far as it could serve the general education of the 
 mind ; they were subjected to such a control and mode 
 of life, as was formerly considered to be good for young 
 priests. They lived, as they still live, in colleges, under 
 
ON ACADEMIC FREEDOM IN GERMAN UNIVERSITIES. 243 
 
 the superintendence of a number of older graduate mem- 
 bers (Fellows) of the College ; in other respects in the 
 style and habits of the well-to-do classes in England. 
 
 The range and the method of the instruction is a 
 more highly developed gymnasial instruction ; though 
 in its limitation to what is afterwards required in the 
 examination, and in the minute study of the contents 
 of prescribed text-books, it is more like the Repeti- 
 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 
 
 R 2 
 
244 ON ACADEMIC FKEEDOM IN GEBMAN UNIVERSITIES. 
 
 students. They need not even live in the University 
 Town, but may spend their stipends where they like, 
 and in many cases may retain the fellowships for an 
 indefinite period. With some exceptions, they only lose 
 it in case they marry, or are elected to certain offices. 
 They are the real successors of the old corporation 
 of students, by and for which the University was 
 founded and endowed. But however beautiful this 
 plan may seem, and notwithstanding the enormous 
 sums devoted to it, in the opinion of all unprejudiced 
 Englishmen it does but little for science ; manifestly 
 because most of these young men, although they are 
 the pick of the students, and in the most favourable 
 conditions possible for scientific work, have in their 
 student-career not come sufficiently in contact with 
 the living spirit of inquiry, to work on afterwards on 
 their own account, and with their own enthusiasm. 
 
 In certain respects the English Universities do 
 a great deal. They bring up their students as cul- 
 tivated men, who are expected not to break through 
 the restrictions of their political and ecclesiastical 
 party, and, in fact, do not thus break through. In 
 two respects we might well endeavour to imitate 
 them. In the first place, together with a lively feeling 
 for the beauty and youthful freshness of antiquity, 
 they develop in a high degree a sense for delicacy 
 and precision in writing which shows itself in the 
 
ON ACADEMIC FREEDOM IN GERMAN UNIVERSITIES. 245 
 
 way in which they handle their mother-tongue. I 
 fear that one of the weakest sides in the instruction 
 of German youth is in this direction. In the second 
 place the English Universities, like their schools, take 
 greater care of the bodily health of their students. 
 They live and work in airy, spacious buildings, sur- 
 rounded by lawns and groves of trees ; they find much 
 of their pleasure in games which excite a passionate 
 rivalry in the development of bodily energy and skill, 
 and which in this respect are far more efficacious 
 than our gymnastic and fencing exercises. It must 
 not be forgotten that the more young men are cut off 
 from fresh air and from the opportunity of vigorous 
 exercise, the more induced will they be to seek an 
 apparent refreshment in the misuse of tobacco and of 
 intoxicating drinks. It must also be admitted that 
 the English Universities accustom their students to 
 energetic and accurate work, and keep them up to 
 the habits of educated society. The moral effect of the 
 more rigorous control is said to be rather illusory. 
 
 The Scotch Universities and some smaller English 
 foundations of more recent origin University College 
 and King's College in London, and Owens College in 
 Manchester are constituted more on the German and 
 Dutch model. 
 
 The development of French Universities has been 
 quite different, and indeed almost in the opposite 
 
246 ON ACADEMIC FREEDOM IN GERMAN UNIVERSITIES. 
 
 direction. In 'accordance with the tendency of the 
 French to throw overboard everything of historic de- 
 velopment to suit some rationalistic theory, their 
 faculties have logically become purely institutes for 
 instruction special schools, with definite regulations 
 for the course of instruction, developed and quite dis- 
 tinct from those institutions which are to further the 
 progress of science, such as the College de France, the 
 Jardin des Plantes, and the Ecole des Etudes Su- 
 perieures. The faculties are entirely separated from 
 one another, even when they are in the same town. 
 The course of study is definitely prescribed, and is 
 controlled by frequent examinations. French teaching 
 is confined to that which is clearly established, and 
 transmits this in a well-arranged, well worked-out 
 manner, which is easily intelligible, and does not ex- 
 cite doubt nor the necessity for deeper inquiry. The 
 teachers need only possess good receptive talents. 
 Thus in France it is looked upon as a false step when 
 a young man of promising talent takes a professorship 
 in a faculty in the provinces. The method of instruc- 
 tion in France is well adapted to give pupils, of even 
 moderate capacity, sufficient knowledge for the routine 
 of their calling. They have no choice between different 
 teachers, and they swear in verba magistri ; this gives 
 a happy self-satisfaction and freedom from doubts. If 
 the teacher has been well chosen, this is sufficient in 
 
ON ACADEMIC FREEDOM IN GERMAN UNIVERSITIES. 247 
 
 ordinary cases, in which the pupil does what he has 
 seen his teacher do. It is only unusual cases that test 
 how much actual insight and judgment the pupil has 
 acquired. The French people are moreover gifted, 
 vivacious, and ambitious, and this corrects many de- 
 fects in their system of teaching. 
 
 A special feature in the organisation of French 
 Universities consists in the fact that the position of 
 the teacher is quite independent of the favour of his 
 hearers ; the pupils who belong to his faculty are 
 generally compelled to attend his lectures, and the far 
 from inconsiderable fees which they pay flow into the 
 chest of the Minister of Education ; the regular salaries 
 of the University professors are defrayed from this 
 source ; the State gives but an insignificant contri- 
 bution towards the maintenance of the University. 
 When, therefore, the teacher has no real pleasure in 
 teaching, or is not ambitious of having a number of 
 pupils, he very soon becomes indifferent to the success 
 of his teaching, and is inclined to take things easily. 
 
 Outside the lecture-rooms, the French students 
 live without control, and associate with young men of 
 other callings, without any special esprit de corps or 
 common feeling. 
 
 The development of the German Universities differs 
 characteristically from these two extremes. Too poor 
 in their own possessions not to be compelled, with 
 
248 ON ACADEMIC FREEDOM IN GERMAN UNIVERSITIES. 
 
 increasing demands for the means of instruction, eagerly 
 to accept the help of the State, and too weak to re- 
 sist encroachments upon their ancient rights in times 
 in which modern States attempt to consolidate them- 
 selves, the German Universities have had to submit 
 themselves to the controlling influence of the State. 
 Owing to this latter circumstance the decision in all 
 important University matters has in principle been 
 transferred to the State, and in times of religious or 
 political excitement this supreme power has occasionally 
 been unscrupulously exerted. But in most cases the 
 States which were working out their own independence 
 were favourably disposed towards the Universities; 
 they required intelligent officials, and the fame of their 
 country's University conferred a certain lustre upon the 
 Government. The ruling officials were, moreover, for 
 the most part students of the University; they re- 
 mained attached to it. It is very remarkable how 
 among wars and political changes in the States fight- 
 ing with the decaying Empire for the consolidation of 
 their young sovereignties, while almost all other privi- 
 leged orders were destroyed, the Universities of Germany 
 saved a far greater nucleus of their internal freedom 
 and of the most valuable side of this freedom, than in 
 conscientious Conservative England, and than in France 
 with its wild chase after freedom. 
 
 We have retained the old conception of students, as 
 
ON ACADEMIC FREEDOM IN GERMAN UNIVERSITIES. 249 
 
 that of young men responsible to themselves, striving 
 after science of their own free will, and to whom it is 
 left to arrange their own plan of studies as they think 
 best. If attendance on particular lectures was enjoined 
 for certain callings what are called ' compulsory lec- 
 tures ' these regulations were not made by the Univer- 
 sity, but by the State, which was afterwards to admit 
 candidates to these callings. At the same time the 
 students had, and still have, perfect freedom to migrate 
 from one German University to another, from Dorpat 
 to Zurich, from Vienna to Gratz ; and in each University 
 they had free choice among the teachers of the same 
 subject, without reference to their position as ordinary 
 or extraordinary professors or as private docents. The 
 students are, in fact, free to acquire any part of their 
 instruction from books ; it is highly desirable that the 
 works of great men of past times should form an essen- 
 tial part of study. 
 
 Outside the University there is no control over the 
 proceedings of the students, so long as they do not 
 come in collision with the guardians of public order. 
 Beyond these cases the only control to which they are 
 subject is that of* their colleagues, which prevents 
 them from doing anything which is repugnant to the 
 feeling of honour of their own body. The Universities 
 of the Middle Ages formed definite close corporations, 
 with their own jurisdiction, which extended to the 
 
250 ON ACADEMIC FKEEDOM IN GEKMAN TJNIVEKSITIES. 
 
 right over life and death of their own members. As they 
 lived for the most part on foreign soil, it was necessary 
 to have their own jurisdiction, partly to protect the 
 members from the caprices of foreign judges, partly to 
 keep up that degree of respect and order, within the 
 society, which was necessary to secure the continuation 
 of the rights of hospitality on a foreign soil; and 
 partly, again, to settle disputes among the members. 
 In modern times the remains of this academic juris- 
 diction have by degrees been completely transferred 
 to the ordinary courts, or will be so transferred ; but it 
 is still necessary to maintain certain restrictions on a 
 union of strong and spirited young men, which guar- 
 antee the peace of their fellow-students and that of 
 the citizens. In cases of collision this is the object of 
 the disciplinary power of the University authorities. 
 This object, however, must be mainly attained by the 
 sense of honour of the students ; and it must be con- 
 sidered fortunate that German students have retained 
 a vivid sense of corporate union, and of what is inti- 
 mately connected therewith, a requirement of honour- 
 able behaviour in the individual. I am by no means 
 prepared to defend every individual regulation in the 
 Codex of Students' Honour; there are many Middle 
 Age remains among them which were better swept 
 away; but that can only be done by the students 
 themselves. 
 
ON ACADEMIC FREEDOM IN GERMAN UNIVERSITIES. 251 
 
 For most foreigners the uncontrolled freedom of 
 German students is a subject of astonishment; the 
 more so as it is usually some obvious excrescences 
 of this freedom which first meet their eyes ; they are 
 unable to understand how young men can be so left 
 to themselves without the greatest detriment. The 
 German looks back to his student life as to his golden 
 age ; our literature and our poetry are full of expres- 
 sions of this feeling. Nothing of this kind is but 
 even faintly suggested in the literature of other Euro- 
 pean peoples. The German student alone has this 
 perfect joy in the time, in which, in the first delight in 
 youthful responsibility, and freed more immediately 
 from having to work for extraneous interests, he can 
 devote himself to the task of striving after the best and 
 noblest which the human race has hitherto been able to 
 attain in knowledge and in speculation, closely joined 
 in friendly rivalry with a large body of associates of 
 similar aspirations, and in daily mental intercourse 
 with teachers from whom he learns something of the 
 workings of the thoughts of independent minds. 
 
 When I think of my own student life, and of the 
 impression which a man like Johannes Miiller, the 
 physiologist, made upon us, I must place a very high 
 value upon this latter point. Anyone who has once 
 come in contact with one or more men of the first rank 
 must have had his whole mental standard altered for 
 
252 ON ACADEMIC FREEDOM IN GERMAN UNIVERSITIES. 
 
 the rest of his life. Such intercourse is, moreover, the 
 most interesting that life can offer. 
 
 You, my younger friends, have received in this 
 freedom of the German students a costly and valuable 
 inheritance of preceding generations. Keep it and 
 hand it on to coming races, purified and ennobled 
 if possible. You have to maintain it, by each, in his 
 place, taking care that the body of German students is 
 worthy of the confidence which has hitherto accorded 
 such a measure of freedom. But freedom necessarily 
 implies responsibility. It is as injurious a present for 
 weak, as it is valuable for strong characters. Do not 
 wonder if parents and statesmen sometimes urge that a 
 more rigid system of supervision and control, like that of 
 the English, shall be introduced even among us. There 
 is no doubt that, by such a system, many a one would 
 be saved who is ruined by freedom. But the State and 
 the Nation is best served by those who can bear free- 
 dom, and have shown that they know how to work and 
 to struggle, from their own force and insight and from 
 their own interest in science. 
 
 My having previously dwelt on the influence of 
 mental intercourse with distinguished men, leads me 
 to discuss another point in which German Universities 
 are distinguished from the English and French ones. 
 It is that we start with the object of having instruc- 
 tion given, if possible, only by teachers who have proved 
 
ON ACADEMIC FREEDOM IN GERMAN UNIVERSITIES. 253 
 
 their own power of advancing science. This also is a 
 point in respect to which the English and French often 
 express their surprise. They lay more weight than the 
 Germans on what is called the ' talent for teaching ' 
 that is, the power of explaining the subjects of instruc- 
 tion in a well-arranged and clear manner, and, if pos- 
 sible, with eloquence, and so as to entertain and to 
 fix the attention. Lectures of eloquent orators at the 
 College de France, Jardin des Plantes, as well as in 
 Oxford and Cambridge, are often the centres of the 
 elegant and the educated world. In Grermany we are 
 not only indifferent to, but even distrustful of, oratorical 
 ornament, and often enough are more negligent than 
 we should be of the outer forms of the lecture. There 
 can be no doubt that a good lecture can be followed 
 with far less exertion than a bad one ; that the matter 
 of the first can be more certainly and completely ap- 
 prehended ; that a well-arranged explanation, which 
 develops the salient points and the divisions of the sub- 
 ject, and which brings it, as it were, almost intuitively 
 before us, can impart, more information in the same 
 time than one which has the opposite qualities. I am 
 by no means prepared to defend what is, frequently, our 
 too great contempt for form in speech and in writing. 
 It cannot also be doubted that many original men, 
 who have done considerable scientific work, have often 
 an uncouth, heavy, and hesitating delivery. Yet I have 
 
254 ON ACADEMIC FKEEDOM IN GEEMAN UNIVERSITIES. 
 
 not infrequently seen that such teachers had crowded 
 lecture-rooms, while empty-headed orators excited 
 astonishment in the first lecture, fatigue in the 
 second, and were deserted in the third. Anyone 
 who desires to give his hearers a perfect conviction 
 of the truth of his principles must, first of all, know 
 from his own experience how conviction is acquired and 
 how not. He must have known how to acquire con- 
 viction where no predecessor had been before him 
 that is, he must have worked at the confines of human 
 knowledge, and have conquered for it new regions. A 
 teacher who retails convictions which are foreign to 
 him, is sufficient for those pupils who depend upon 
 authority as the source of their knowledge, but not for 
 such as require bases for their conviction which extend 
 to the very bottom. 
 
 You will see that this is an honourable confidence 
 which the nation reposes in you. Definite courses 
 and specified teachers are not prescribed to you. You 
 are regarded as men whose unfettered conviction is 
 to be gained ; who know how to distinguish what 
 is essential from what is only apparent; who can no 
 longer be appeased by an appeal to any authority, and 
 who no longer let themselves be so appeased. Care is 
 also always taken that you yourselves should penetrate 
 to the sources of knowledge in so far as these consist 
 
ON ACADEMIC FREEDOM IN GERMAN UNIVERSITIES. 255 
 
 in books and monuments, or in experiments, and in the 
 observation of natural objects and processes. 
 
 Even the smaller German Universities have their 
 own libraries, collections of casts, and the like. And 
 in the establishment of laboratories for chemistry, 
 microscopy, physiology, and physics, Germany has 
 preceded all other European countries, who are now be- 
 ginning to emulate her. In our own University we may 
 in the next few weeks expect the opening of two new 
 institutions devoted to instruction in natural science. 
 
 The free conviction of the student can only be 
 acquired when freedom of expression is guaranteed to 
 the teacher's own conviction the liberty of teaching. 
 This has not always been ensured, either in Germany 
 or in the adjacent countries. In times of political and 
 ecclesiastical struggle the ruling parties have often 
 enough allowed themselves to encroach ; this has 
 always been regarded by the German nation as an 
 attack upon their sanctuary. The advanced political 
 freedom of the new German Empire has brought a 
 cure for this. At this moment, the most extreme con- 
 sequences of materialistic metaphysics, the boldest 
 speculations upon the basis of Darwin's theory of evo- 
 lution, may be taught in German Universities with as 
 little restraint as the most extreme deification of Papal 
 Infallibility. As in the tribune of European Parlia- 
 
256 ON ACADEMIC FREEDOM IN GERMAN UNIVERSITIES. 
 
 ments it is forbidden to suspect motives or indulge in 
 abuse of the personal qualities of our opponents, so 
 also is any incitement to such acts as are legally for- 
 bidden. But there is no obstacle to the discussion of 
 a scientific question in a scientific spirit. In English 
 and French Universities there is less idea of liberty of 
 teaching in this sense. Even in the College de France 
 the lectures of a man of Kenan's scientific impor- 
 tance and earnestness are forbidden. 
 
 I have to speak of another aspect of our liberty of 
 teaching. That is, the extended sense in which Ger- 
 man Universities have admitted teachers. In the 
 original meaning of the word, a doctor is a e teacher,' or 
 one whose capacity as teacher is recognised. In the 
 Universities of the Middle Ages any doctor who found 
 pupils could set up as teacher. In course of time the 
 practical signification of the title was changed. Most 
 of those who sought the title did not intend to act as 
 teachers, but only needed it as an official recognition 
 of their scientific training. Only in Germany are 
 there any remains of this ancient right. In accord- 
 ance with the altered meaning of the title of doctor, 
 and the minuter specialisation of the subjects of in- 
 struction, a special proof of more profound scientific 
 proficiency, in the particular branch in which they wish 
 to habilitate, is required from those doctors who desire 
 to exercise the right of teaching. In most German 
 
ON ACADEMIC FREEDOM IN GERMAN UNIVERSITIES. 257 
 
 Universities, moreover, the legal status of these habili- 
 tated doctors as teachers is exactly the same as that of 
 the ordinary professors. In a few places they are 
 subject to some slight restrictions which, however, 
 have scarcely any practical effect. The senior teachers 
 of the University, especially the ordinary professors, 
 have this amount of favour, that, on the one hand, in 
 those branches in which special apparatus is needed 
 for instruction, they can more freely dispose of the 
 means belonging to the State ; while on the other it 
 falls to them to hold the examinations in the faculty, 
 and, as a matter of fact, often also the State examina- 
 tion. This naturally exerts a certain pressure on the 
 weaker minds among the students. The influence of 
 examinations is, however, often exaggerated. In the 
 frequent migrations of our students, a great number 
 of examinations are held in which the candidates have 
 never attended the lectures of the examiners. 
 
 On no feature of our University arrangements do 
 foreigners express their astonishment so much as about 
 the position of private docents. 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 
 H. s 
 
258 ON ACADEMIC FBEEDOM IN GEKMAN UNIVERSITIES. 
 
 equally surprised that the faculties so readily admit 
 young inen who at any moment may change from 
 assistants to competitors; and further, that only in 
 the most exceptional cases is anything ever heard of 
 unworthy means of competition in what is a matter of 
 some delicacy. 
 
 The appointment to vacant professorships, like the 
 admission of private docents, rests, though not uncon- 
 ditionally, and not in the last resort, with the faculty, 
 that is with the body of ordinary professors. These 
 form, in German Universities, that residuum of former 
 colleges of doctors to which the rights of the old 
 corporations have been transferred. They form as it 
 were a select committee of the graduates of a former 
 epoch, but established with the co-operation of the 
 Government. The usual form for the nomination of 
 new ordinary professors is that the faculty proposes 
 three candidates to Government for its choice ; where 
 the Government, however, does not consider itself 
 restricted to the candidates proposed. Excepting in 
 times of heated party conflict it is very unusual for the 
 proposals of the faculty to be passed over. If there is 
 not a very obvious reason for hesitation it is always a 
 serious personal responsibility for the executive officials 
 to elect, in opposition to the proposals of competent 
 judges, a teacher who has publicly to prove his 
 capacity before large circles. 
 
ON ACADEMIC FREEDOM IN GERMAN UNIVERSITIES. 
 
 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 number 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 
 
 s 2 
 
260 ON ACADEMIC FREEDOM IN GERMAN UNIVERSITIES. 
 
 there are two to teach one and the same branch ; and 
 even if there is no difference in the official definition 
 of the subject, there will be a difference in the scien- 
 tific tendencies of the teachers ; they will be able to 
 divide the work in such a manner that each has that 
 side which he most completely masters. Two distin- 
 guished teachers who are thus complementary to each 
 other, form then so strong a centre of attraction for 
 the students that both suffer no loss of hearers, though 
 they may have to share among themselves a number of 
 the less zealous ones. 
 
 The disagreeable effects of rivalry will be feared 
 by a teacher who does not feel quite certain in his 
 scientific position. This can have no considerable 
 influence on the official decisions of the faculty when 
 it is only a question of one, or of a small number, of 
 the voters. 
 
 The predominance of a distinct scientific school in 
 a faculty may become more injurious than such per- 
 sonal interests. When the school has scientifically out- 
 lived itself, students will probably migrate by degrees 
 to other Universities. This may extend over a long 
 period, and the faculty in question will suffer during 
 that time. 
 
 We see best how strenuously the Universities under 
 this system have sought to attract the scientific ability 
 of Germany when we consider how many pioneers have 
 
ON ACADEMIC FEEEDOM IN GERMAN UNIVERSITIES. 261 
 
 remained outside the Universities. The answer to such 
 an inquiry is given in the not infrequent jest or sneer 
 that all wisdom in Germany is professorial wisdom. If 
 we look at England, we see men like Humphry Davy, 
 Faraday, Mill, Grote, who have had no connection with 
 English Universities. If, on the other hand, we deduct 
 from the list of German men of science those who, 
 like David Strauss, have been driven away by Govern- 
 ment for ecclesiastical or for political reasons, and those 
 who, as members of learned Academies, had the right 
 to deliver lectures in the Universities, as Alexander and 
 Wilhelm 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 Koyal Institution of 
 London, a private Society, which provides for its mem- 
 bers and others short courses of lectures on the Progress 
 of Natural Science, should have been able to retain 
 permanently the services of men of such scientific 
 importance as Humphry Davy and Faraday. It was 
 no question of great emoluments; these men were 
 manifestly attracted by a select public consisting of 
 men and women of independent mental culture. In 
 Germany the Universities are unmistakably the insti- 
 tutions which exert the most powerful attraction on 
 
262 ON ACADEMIC FEEEDOM IN GEEMAN UNIVEESITIE& 
 
 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 
 Komanic 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 
 
ON ACADEMIC FKEEDOM IN GEKMAN UNIVEESITIES. 263 
 
 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 
 soon be dispirited in his work if he could not count 
 upon the zeal and the interest of his hearers. The 
 
264 ON ACADEMIC FREEDOM IN GERMAN UNIVERSITIES. 
 
 estimate which he places on his task will depend on 
 how far he is followed by the appreciation of a suffi- 
 cient number of, at any rate, his more intelligent 
 hearers. The influx of hearers to the lectures of a 
 teacher has no slight influence upon his fame and 
 promotion, and, therefore, upon the composition of 
 the body of teachers. In all these respects, it is 
 assumed that the general public opinion among the 
 students cannot go permanently wrong. The majority 
 of them who are, as it were, the representatives of 
 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 students 
 
ON ACADEMIC FREEDOM IN GERMAN UNIVERSITIES. 265 
 
 must not be allowed to sink. If that were the case, 
 the dangers of academic freedom would choke its 
 blessings. It must therefore not be looked upon as 
 pedantry, or arrogance, if the Universities are scrupu- 
 lous in the admission of students of a different style 
 of education. It would be still more dangerous if, 
 for any extraneous reasons, teachers were introduced 
 into the faculty, who have not the complete qualifica- 
 tions of an independent academical teacher. 
 
 Do not forget, my dear colleagues, that you are 
 in a responsible position. You have to preserve the 
 noble inheritance of which I have spoken, not only 
 for your own people, but also as a model to the 
 widest circles of humanity. You will show that 
 youth also is enthusiastic, and will work for inde- 
 pendence of conviction. I say work; for indepen- 
 dence of conviction is not the facile assumption of ; 
 untested hypotheses, but can only be acquired as 
 the fruit of conscientious inquiry and strenuous 
 labour. You must show that a conviction which 
 you yourselves have worked out is a more 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. 
 
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