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diff --git a/.gitattributes b/.gitattributes new file mode 100644 index 0000000..6833f05 --- /dev/null +++ b/.gitattributes @@ -0,0 +1,3 @@ +* text=auto +*.txt text +*.md text diff --git a/77725-0.txt b/77725-0.txt new file mode 100644 index 0000000..53053ef --- /dev/null +++ b/77725-0.txt @@ -0,0 +1,6664 @@ +*** START OF THE PROJECT GUTENBERG EBOOK 77725 *** + + + + +Transcriber’s Notes: + + Underscores _ before and after a word or phrase indicate _italics_ + in the original text. + Equal signs = before and after a word or phrase indicate =bold= + in the original text. + Small capitals have been converted to SOLID capitals. + Illustrations have been moved so they do not break up paragraphs. + Deprecated spellings have been preserved. + Typographical and punctuation errors have been silently corrected. + The list of works by John Tyndall has been moved from the front to the + end of the file. + + + + + POPULAR LECTURES + ON + SCIENTIFIC SUBJECTS + + BY + HERMANN VON HELMHOLTZ + + TRANSLATED BY + E. ATKINSON, PH.D., F.C.S. + FORMERLY PROFESSOR OF EXPERIMENTAL SCIENCE, STAFF COLLEGE + + SECOND SERIES + + _WITH AN AUTOBIOGRAPHY OF THE AUTHOR_ + + [Illustration] + + LONGMANS, GREEN, AND CO. + 39 PATERNOSTER ROW, LONDON + NEW YORK, BOMBAY, AND CALCUTTA + 1908 + + _All rights reserved_ + + _BIBLIOGRAPHICAL NOTE._ + + _Issued in Silver Library, March 1893; Reprinted + July 1895, May 1898, June 1900, December 1903, + August 1908._ + + + + +PREFACE. + + +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 Professor G. Croome Robertson, +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 +exception has been taken. These statements were a fair representation +of the impression produced on the mind of a foreigner by a state of +things which no longer exists in those Universities, at least to the +same extent. The reform in the University system, which may be said to +date from the year 1854, has brought about so many alterations both in +the form and in the spirit of the regulations, that older members of +the University have been known to speak of the place as so changed that +they could scarcely recognise it. Hence, in respect of this article, +I have availed myself of the liberty granted by Professor Helmholtz, +and have altogether omitted some passages, and have slightly modified +others, which would convey an erroneous 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 reproduce the original; nor have I in any such +cases allowed a regard for form to interfere with the plain duty of +exactly rendering the author’s meaning. + + E. ATKINSON. + + PORTESBERY HILL, CAMBERLEY: + _Dec. 1880_. + + + + + CONTENTS. + + LECTURE PAGE + I. GUSTAV MAGNUS. IN MEMORIAM 1 + II. ON THE ORIGIN AND SIGNIFICANCE OF GEOMETRICAL AXIOMS 27 + III. ON THE RELATION OF OPTICS TO PAINTING 73 + I. Form 78 + II. Shade 94 + III. Colour 110 + IV. Harmony of Colour 124 + IV. ON THE ORIGIN OF THE PLANETARY SYSTEM 139 + V. ON THOUGHT IN MEDICINE 199 + VI. ON ACADEMIC FREEDOM IN GERMAN UNIVERSITIES 237 + VII. HERMANN VON HELMHOLTZ. AN AUTOBIOGRAPHICAL SKETCH 266 + + + + +GUSTAV MAGNUS. + +_In Memoriam._ + + +_Address delivered in the Leibnitz meeting of the Academy of Sciences +on July 6, 1871._ + +The honourable duty has fallen on me of expressing in the name of +this Academy what it has lost in Gustav Magnus, who belonged to it +for thirty years. As a grateful pupil, as a friend, and finally as +his successor, it was a pleasure to me as well as a duty to fulfil +such a task. But I find the best part of my work already done by our +colleague Hofmann at the request of the German Chemical Society, of +which he is the President. He has solved the difficulty of giving a +picture of the life and work of Magnus in the most complete 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 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 restricted. 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. Heinrich Gustav Magnus was born in Berlin on May 2, 1802, +the fourth of six brothers, who by their talents have distinguished +themselves in various directions. The father, Johann Matthias, was +chief of a wealthy commercial house, whose first concern was to secure +to his children a free development of their individual capacity +and inclinations. Our departed friend showed very early a greater +inclination for the study of mathematics and natural philosophy than +for that of languages. His father arranged his instruction accordingly, +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 entirely 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 Dulong, Thénard 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 professor in 1834, and ordinary +professor in 1845, and so distinguished himself by his scientific +labours at this time, that nine years after his habilitation, on +January 27, 1840, he was elected a member of this Academy. From 1832 +until 1840 he taught physics in the Artillery and Engineering School; +and from 1850 until 1856 chemical technology in the Gewerbe Institut. +For a long time he gave the lectures in his own house, using his own +instruments, which gradually developed into the most splendid physical +collection in existence at that time, and which the State afterwards +purchased for the University. His lectures were afterwards given in the +University, and he only retained the laboratory in his own house for +his own private work and for that of his pupils. + +His life was passed thus in quiet but unremitting activity; travels, +sometimes for scientific or technical studies, sometimes also in the +service of the State, and occasionally for recreation, interrupted his +work here from time to time. His experience and knowledge of business +were often in demand by the State on various commissions; among +these may be especially mentioned the part he took in the chemical +deliberations of the Agricultural Board (_Landes-Economie Collegium_), +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 during +March he was scarcely able to leave his bed, and he died on April 4. + +[Footnote 1: Carcinoma recti.] + +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 mortals. The harmonious tendency and +cultivation of his mind could be recognised in the natural grace of his +behaviour, in the cheerfulness and firmness of his disposition, in the +warm amiability of his intercourse with others. There was in all this, +much more than the mere acquisition of the outer forms of politeness +can ever reach, where they are not illuminated by a warm sympathy and +by a fine feeling for the beautiful. + +Accustomed from an early age to the regulated and prudent activity of +the commercial house in which he grew up, he retained that skill in +business which he had so frequently to exercise in the administration +of the affairs of this Academy, of the philosophical faculty, and +of the various Government commissions. He retained from thence the +love of order, the tendency towards the actual, and towards what is +practically attainable, even although the chief aim of his activity +was an ideal one. He understood that the pleasant enjoyment of an +existence free from care, and intercourse with the most amiable circle +of relatives and friends, do not bring a lasting satisfaction; but +work only, and unselfish work for an ideal aim. Thus he laboured, not +for the increase of riches, but for science; not as a dilettante and +capriciously, but according to a fixed aim and indefatigably; not in +vanity, catching at striking discoveries, which might at once have made +his name celebrated. He was, on the contrary, a master of faithful, +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 simultaneously, 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 characteristic 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 appliances of his private +laboratory. This was the way in which I was brought in close relation +to him, when I found myself in Berlin for the purpose of passing the +Government medical examination. + +He invited me at that time (I myself would not have ventured to propose +it) to extend my experiments on fermentation and putrefaction in new +directions, and to apply other methods, which required greater means +than a young army surgeon living on his pay could provide himself with. +At that time I worked with him almost daily for about three months, +and thus gained a deep and lasting impression of his goodness, his +unselfishness, and his perfect freedom from scientific jealousy. + +By such a course he not only surrendered the external 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 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 apparatus 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 recollect very well with what wonder and +admiration we students saw him experiment, not merely because all the +experiments were successful and brilliant, but because they scarcely +seemed to occupy or to disturb his thoughts. The easy and clear flow +of his discourse went on without interruption; each experiment came in +its right place, was performed quickly, without haste or hesitation, +and was then put aside. + +I have already mentioned that the valuable collection of apparatus came +into the possession of the University during his lifetime. He specially +wished that what he had collected and constructed as appliances 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 +laboratory, 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, 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 possession, 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 revolutionising +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 hesitation Germans applied +themselves to the study of natural science in this century, whilst +they had taken so prominent 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 principle to their +ultimate sources, and to trouble ourselves but little about what has no +connection with fundamental principles, and especially about practical +consequences 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, +politically 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 intellectual life, for all this there was no time during the +storm of war; each man had to hold to his own party, every incipient +change of opinion was looked upon as treachery, and excited bitter +wrath. Owing to the Reformation, intellectual life had lost its old +stability and cohesion; everything appeared in a new light, and new +questions arose. The German mind could not be quieted with outward +uniformity; when it was not convinced and satisfied, it did not allow +its doubt to remain silent. Thus it was theology, and next to it +classical philology and philosophy, which, partly as scientific aids +of theology, partly for what they could do for the solution of the +new moral, æsthetical, and metaphysical problems, laid claim almost +exclusively to the interest of scientific culture. Hence it is clear +why the Protestant nations, as well as that part of the Catholics +which, wavering in its old faith, only remained outwardly in connection +with its church, threw itself with such zeal on philosophy. Ethical +and metaphysical problems 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 enumerate the actual results +which the last century gained by this work. It excited soaring hopes, +and it cannot be denied that metaphysics has a dangerous attraction +for the German mind; it could not again abandon it until all its +hiding-places had been searched through and it had satisfied itself +that for the present nothing more is to be found there. + +Then, in the second half of the last century, the rejuvenescent +intellectual life of the nation began to cultivate its artistic +flowers; the clumsy language transformed itself into one of the most +expressive instruments of the human mind; out of what was still the +hard, poor, and wearisome condition of civil and political life, the +results of the religious war, in which the figure of the Prussian +hero-king only now cast the first hope of a better future, to be again +followed by the misery of the Napoleonic war,--out of this joyless +existence, all sensitive minds gladly fled into the flowery land opened +out by German poetry, rivalling as it did the best poetry of all times +and of all peoples; or in the sublime aspects of philosophy they +endeavoured to sink reality in oblivion. + +And the natural sciences were on the side of this real world, so +willingly overlooked. Astronomy alone could at that time offer great +and sublime prospects; in all other branches long and patient labour +was still necessary before great principles could be attained; before +these subjects could have a voice in the great questions of human +life; or before they became the powerful means of the authority of man +over the forces of nature which they have since become. The labour of +the natural philosopher seems narrow, low, and insignificant compared +with the great conceptions of the philosopher and of the poet; it was +only those natural philosophers who, like Oken, rejoiced in poetical +philosophical conceptions, who found willing auditors. + +Far be it from me as a one-sided advocate of scientific interests to +blame this period of enthusiastic excitement; 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 resource 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 corresponded at first to +the natural soaring of a fresh youthful ambition, but which afterwards, +in the age of the Epigones of the Romantic school and of the philosophy +of Identity, fell into sentimental straining after sublimity and +inspiration, a reaction took place, and was carried out not merely in +the regions of science, but also in history, in art, and in philology. +In the last departments, too, where we deal directly with products of +activity of the human mind, and where, therefore, a construction _à +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, +before 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 endeavoured 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 continually preached +to his pupils that no reasoning, however plausible it might seem, +avails against actual fact, and that observation and experiment must +decide; and he was always anxious that every practicable experiment +should be made which could give practical confirmation or refutation +of an assumed law. He did not limit in any way the applicability of +scientific methods in the investigation of inanimate nature, but in his +research on the gases of the blood (1837) he dealt a blow at the heart +of vitalistic theories. He led physics to the centre of organic change, +by laying a scientific foundation for a correct theory of respiration; +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 mathematical 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 standpoint +which science has now attained, it must be confessed that his +distrust of the mathematical physics of that date was not unfounded. +At that time no separation 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 elements which formed the basis +of calculation was put forth as axioms of metaphysical necessity, +and postulated a similar kind of necessity for the results. I need +only recall to you the great part which hypotheses as to the atomic +structure of bodies played in mathematical physics during the first +half of this century, whilst as good as nothing was known of atoms; +and, for instance, hardly anything was known of the extraordinary +influence which heat has on molecular 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 assumption 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 mathematicians who in England followed +Faraday’s lead, Stokes, W. Thomson, and Clerk-Maxwell. It is now +understood that mathematical physics is a purely experimental science; +that it has no other principles to follow than those of experimental +physics. In our immediate experience we find bodies variously formed +and constituted; only with such can we make our observations and +experiments. Their actions are made up of the actions which each of +their parts contributes to the sum of the whole; and hence, if we wish +to know the simplest and most general law of the action of masses and +substances found in nature upon one another, and if we wish to divest +these laws of the accidents of form, magnitude, and position of the +bodies concerned, we must go back to the laws 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 experimentally, 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 mathematical +physics only investigates the laws of action of the elements of a body +independently of the accidents of form, in a purely empirical manner, +and is therefore 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 continues 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 antagonism. 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 experimentalise profitably +who have a clear-sighted knowledge of theory, and know how to propound +and pursue the right questions; and, on the other hand, only those can +theorise with advantage who have great practice in experiments. The +discovery of spectrum analysis is the most brilliant example within our +recollection of such an interpenetration of theoretical knowledge and +experimental skill. + +I am not aware whether Magnus subsequently expressed other views as to +the relation of experimental and mathematical physics. In any case, +those who regard his former desertion of mathematical physics as a +reaction against the misuse of speculation carried too far, must also +admit that in the older mathematical 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 researches 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 +progress. 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 discovered. But just as we all, +like Magnus, are opposed 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 to avoid a law, or a +possible connection between newly discovered facts. + +It must here be mentioned that Faraday, another great physicist, worked +in England exactly in the same direction, and with the same object; +to whom, on that account, Magnus was bound by the heartiest sympathy. +With Faraday, the antagonism to the physical theories hitherto held, +which treated of atoms and forces acting at a distance, was even more +pronounced 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 experiment, 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 deflected 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 aperture through which +they flow, partly in consequence of the manner in which they flow +to it; and how their resolution into drops is caused by external +agitation. He applied the principle of the stroboscopic disk in +observing the phenomena, by looking at the jet through small slits in +a rotating disk. He grouped the various phenomena with peculiar tact, +so that those among them which are alike were easily seen, and one +elucidated the other. And if a final mechanical explanation is not +always attained, yet the reason for a great number of characteristic +features of the individual phenomena is plain. In this respect many of +his researches--I might specially commend those on the efflux of jets +of water--are excellent models of what Goethe theoretically advanced, +and in his physical 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 declared: ‘When I +commenced the experiment just described, I confidently hoped to find +that thermo-electrical currents are due to a motion of heat.’ In this +sense he investigated the cases in which the thermo-electrical circuit +consisted of a single metal in which there were alternately hard +portions, and such as had been softened by heat; or those it 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 conductivity for heat, using this expression in its +ordinary meaning, and he had to content himself with the obviously +imperfect explanation that two pieces of the same metal at different +temperatures acted like dissimilar conductors, which like liquids do +not fall in with the potential series. The solution was first furnished +by the two general laws of the mechanical theory of heat. Magnus’s hope +was not unfulfilled. Sir W. Thomson discovered that alterations in the +conductivity for heat, though such as were produced by the electrical +current itself, were indeed the sources of the current. + +It is the nature of the scientific direction which Magnus pursued in +his researches, that they build many a stone into the great fabric of +science, which give it an ever broader support, and an ever growing +height, without its appearing to a fresh observer as a special and +distinctive work due to the sole exertion of any one scientific man. +If we wish to explain the importance of each stone for its special +place, how difficult to procure it, and how skilfully it was worked, we +must presuppose either that the hearer knows the entire history of the +building, or we must explain it to him, by which more time is lost than +I can now claim. + +Thus it is with Magnus’s researches. Wherever he has attacked, he has +brought out a host of new and often remarkable facts; he has carefully +and accurately observed them, and he has brought them in connection +with the great fabric of science. He has, moreover, bequeathed +to science a great number of ingenious and carefully devised new +methods, as instruments with which future generations will continue to +discover hidden veins of the noble metal of everlasting laws in the +apparently waste and wild chaos of accident. Magnus’s name will always +be mentioned in the first line of those on whose labours the proud +edifice of the science of Nature reposes; of the science which has so +thoroughly remodelled the life of modern humanity by its intellectual +influence, as well as by its having subjugated 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 physical conditions; he was afterwards +exclusively a physicist. But parallel with this he cultivated a very +extended study of technology, which of itself would alone have occupied +a man’s life. + +He has departed, after a rich life and a fruitful activity. The old +law that no man’s life is free from pain must have been applied to him +also; and yet his life seems to have been especially happy. He had +what men generally desire most; but he knew how to ennoble external +fortune by putting it at the service of unselfish objects. To him was +granted, what is dearest to the mind of a noble spirit, to dwell in +the centre of an affectionate family, and in a circle of faithful and +distinguished friends. But I count his rarest happiness 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 intellectual 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 satisfactory 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 ORIGIN AND SIGNIFICANCE OF GEOMETRICAL AXIOMS. + + +_Lecture delivered in the Docenten Verein in Heidelberg, in the year +1870._ + +The fact that a science can exist and can be developed 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 +inconsistency have so little dared to raise their eyes. It escapes the +tedious and troublesome task of collecting experimental facts, which +is the province of the natural sciences in the strict sense of the +word; the sole form of its scientific method is deduction. Conclusion +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 architecture, +the construction of machinery no less than mathematical physics, are +continually calculating relations of space of the most varied kind +by geometrical principles; they expect that the success of their +constructions and experiments shall agree with these calculations; and +no case is known in which this expectation has been falsified, provided +the calculations were made correctly and with sufficient data. + +Indeed, the fact that geometry exists, and is capable 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 +philosophical systems, that there can be a cognition of principles +destitute of any bases drawn from experience. In the answer to Kant’s +celebrated question, ‘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. 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 peculiarities +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.[2] + +[Footnote 2: In his book, _On the Limits of Philosophy_, Mr. W. Tobias +maintains 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 +dimensions (_Ibid._ part i. sect. i. § 3, p. 41); that only one +straight line is possible between two points (_Ibid._ part ii. sect. +i. ‘On the Axioms of Intuition’), as axioms which express _a priori_ +the conditions of intuition by the senses. It is not here the question, +whether these axioms were originally given as intuition of space, or +whether they are only the starting-points from which the understanding +can develop such axioms _a priori_ on which my critic insists.] + +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 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 +virtually 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 elementary axioms of +geometry, that is, the mathematical doctrine of space, finds on his +path that unbroken chain of conclusions of which I just spoke, by which +the ever more varied and more complicated figures are brought within +the domain of law. But even in their first elements certain principles +are laid down, with respect to which geometry confesses that she cannot +prove them, and can only assume that anyone who understands the essence +of these principles will at once admit their correctness. These are the +so-called axioms. + +For example, the proposition that if the shortest line drawn between +two points is called a _straight_ line, there can be only one such +straight line. Again, it is an axiom that through any three points +in space, not lying in a straight line, a plane may be drawn, i.e. +a surface which will wholly include every straight line joining any +two of its points. Another axiom, about which there has been much +discussion, affirms that through a point lying without a straight +line only one straight line can be drawn parallel to the first; two +straight lines that lie in the same plane and never meet, however far +they may be produced, being called parallel. There are also axioms that +determine the number of dimensions of space and its surfaces, lines and +points, showing how they are continuous; as in the propositions, that +a solid is bounded by a surface, a surface by a line and a line by a +point, that the point is indivisible, that by the movement of a point +a line is described, by that of a line a line or a surface, by that of +a surface a surface or a solid, but by the movement of a solid a solid +and nothing else is described. + +Now what is the origin of such propositions, unquestionably true yet +incapable of proof in a science where everything else is reasoned +conclusion? Are they inherited from the divine source of our reason as +the idealistic philosophers think, or is it only that the ingenuity of +mathematicians has hitherto not been penetrating enough to find the +proof? Every new votary, coming with fresh zeal to geometry, naturally +strives to succeed where all before him have failed. And it is quite +right that each should make the trial afresh; for, as the question has +hitherto stood, it is only by the fruitlessness of one’s own efforts +that one can be convinced of the impossibility of finding a proof. +Meanwhile solitary inquirers are always from time to time appearing who +become so deeply entangled 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 prescribed for the demonstration we +have not involuntarily 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 essential 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 establish the existence of definite figures. They show that +points, straight lines, or circles such as the problem requires to be +constructed are possible under all conditions, 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 supposed 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 translation 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-suppositions we make, in employing +the common constructive 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 submitted to it that is not already contained in the +equations which give occasion for its being applied. The recent +investigations in question have accordingly been conducted almost +exclusively by means of the purely abstract methods of analytical +geometry. + +However, after discovering by the abstract method what are the points +in question, we shall best get a distinct view of them by taking a +region of narrower limits than our own world of space. Let us, as we +logically may, suppose reasoning beings of only two dimensions to live +and move on the surface of some solid body. We will assume that they +have not the power of perceiving anything outside this surface, but +that upon it they have perceptions similar to ours. If such beings +worked out a geometry, they would of course assign only two dimensions +to their space. They would ascertain that a point in moving describes a +line, and that a line in moving describes a surface. But they could as +little represent to themselves what further spatial construction would +be generated by a surface moving out of itself, as we can represent +what would be generated by a solid moving out of the space we know. +By the much-abused expression ‘to represent’ or ‘to be able to think +how something happens’ I understand--and I do not see how anything +else can be understood by it without loss of all meaning--the power +of imagining the whole series of sensible impressions that would be +had in such a case. Now as no sensible impression is known relating +to such an unheard-of event, as the movement to a fourth dimension +would be to us, or as a movement to our third dimension would be to +the inhabitants of a surface, such a ‘representation’ is as impossible +as the ‘representation’ of colours would be to one born blind, if a +description of them in general terms could be given to him. + +Our surface-beings would also be able to draw shortest lines in their +superficial space. These would not necessarily be straight lines in +our sense, but what are technically called _geodetic lines_ of the +surface on which they live; lines such as are described by a _tense_ +thread laid along the surface, and which can slide upon it freely. I +will henceforth speak of such lines as the _straightest_ lines of any +particular surface or given space, so as to bring out their analogy +with the straight line in a plane. I hope by this expression to make +the conception more easy for the apprehension of my non-mathematical +hearers without giving rise to misconception. + +Now if beings of this kind lived on an infinite plane, their geometry +would be exactly the same as our planimetry. They would affirm +that only one straight line is possible between two points; that +through a third point lying without this line only one line can be +drawn parallel to it; that the ends of a straight line never meet +though it is produced to infinity, and so on. Their space might be +infinitely extended, but even if there were limits to their movement +and perception, they would be able to represent to themselves a +continuation beyond these limits; and thus their space would appear +to them infinitely extended, 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 between 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 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. Accordingly, +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 +of all were the same; nor are more examples necessary to show that +geometrical axioms must vary according to the kind of space inhabited +by beings whose powers of reason are quite in conformity with ours. But +let us proceed still farther. + +Let us think of reasoning beings existing on the surface of an +egg-shaped body. Shortest lines could be drawn between three points of +such a surface and a triangle constructed. But if the attempt were made +to construct congruent triangles at different parts of the surface, it +would be found that two triangles, with three pairs of equal sides, +would not have their angles equal. The sum of the angles of a triangle +drawn at the sharper pole of the body would depart farther from two +right angles than if the triangle were drawn at the blunter pole or at +the equator. Hence it appears that not even such a simple figure as +a triangle can be moved on such a surface without change of form. It +would also be found that if circles of equal radii were constructed at +different parts of such a surface (the length of the radii being always +measured by shortest lines along the surface) the periphery would be +greater at the blunter than at the sharper end. + +We see accordingly that, if a surface admits of the figures lying on it +being freely moved without change of any of their lines and angles as +measured along it, the property is a special one and does not belong to +every kind of surface. The condition under which a surface possesses +this important property was pointed out by Gauss in his celebrated +treatise on the curvature of surfaces.[3] 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 everywhere equal over the whole +extent of the surface. + +[Footnote 3: Gauss, _Werke_, Bd. IV. p. 215, first published in +_Commentationes Sec. Reg. Scientt. Gottengensis recentiores_, vol. vi., +1828.] + +Gauss showed at the same time that this measure of curvature is not +changed if the surface is bent without 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 the axiom of parallels does not hold good. +This is a kind of curved surface which is, as it were, geometrically +the counterpart of a sphere, and which has therefore been called the +_pseudospherical surface_ by the distinguished Italian mathematician E. +Beltrami, who has investigated its properties.[4] 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 directions, 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. + +[Footnote 4: _Saggio di Interpretazione della Geometria Non-Euclidea_, +Napoli, 1868.--_Teoria fondamentale degli Spazii di Curvatura costante, +Annali di Matematica_, Ser. II. Tom. II. pp. 232-55. Both have been +translated into French by J. Hoüel, _Annales Scientifiques de l’Ecole +Normale_, Tom V., 1869.] + +Like the plane and the sphere, pseudospherical surfaces have their +measure of curvature constant, so that every piece of them can be +exactly applied to every 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 pseudospherical +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 example, 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 symmetry AB, the two arcs _ab_ will describe a pseudospherical +concave-convex surface like that of the ring. Above and below, towards +_aa_ and _bb_, 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 pseudospherical +surface may be rolled up into the shape of a champagne-glass (Fig. 2), +with tapering stem infinitely prolonged. But the surface is always +necessarily bounded by a sharp edge beyond which it cannot be directly +continued. Only by supposing each single piece of the edge cut loose +and drawn along the surface of the ring or glass, can it be brought to +places of different flexure, at which farther continuation of the piece +is possible. + +In this way too the straightest lines of the pseudospherical 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 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. + +[Illustration: FIG. 1.] + +[Illustration: FIG. 2.] + +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. Lobatchewsky, professor of mathematics at Kasan,[5] 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. + +[Footnote 5: _Principien der Geometrie_, Kasan, 1829-30.] + +Thus we see that in the geometry of two dimensions 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 geometry 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 infinite +continuation of pseudospherical surfaces. Inhabiting, as we do, a +space of three dimensions and endowed with organs of sense for their +perception, we can represent to ourselves the various cases in which +beings on a surface might have to develop their perception 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 analogous 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 measurable, that is, they may be brought +to determination of magnitudes (lines, angles, surfaces, volumes). +Problems in geometry can therefore be solved, by finding methods of +calculation for arriving at unknown magnitudes from known ones. This +is done in _analytical geometry_, where all forms of space are treated +only as quantities and determined by means of other quantities. Even +the axioms themselves make reference to magnitudes. The straight +line is defined as the _shortest_ between two points, which is a +determination of quantity. The axiom of parallels declares that if +two straight lines in a plane do not intersect (are parallel), the +alternate angles, or the corresponding angles, made by a third line +intersecting them, are equal; or it may be laid down instead that the +sum of the angles of any triangle is equal to two right angles. These, +also, are determinations of quantity. + +Now we may start with this view of space, according to which the +position of a point may be determined 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. Riemann of Göttingen.[6] It has the +peculiar advantage that all its operations consist in pure calculation +of quantities, which quite obviates the danger of habitual perceptions +being taken for necessities of thought. + +[Footnote 6: Ueber die Hypothesen welche der Geometrie zu Grunde +liegen, Habilitationsschrift vom 10 Juni 1854. (_Abhandl. der königl. +Gesellsch. zu Göttingen_, Bd. XIII.)] + +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. Riemann calls a system of differences in which one thing can +be determined by _n_ measurements an ‘_n_fold 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 dimensions, inasmuch as each +colour, according to the investigations of Thomas Young and of Clerk +Maxwell,[7] 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. + +[Footnote 7: Helmholtz’s _Popular Lectures_, Series I. p. 243.] + +In the same way we may consider the system of simple tones[8] 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 +between space of three dimensions and other aggregates. We can, as +we know from daily experience, compare the vertical distance of two +points with the horizontal distance of two others, because we can apply +a measure first to the one pair and then to the other. But we cannot +compare the difference between two tones of equal pitch and different +intensity, with that between two tones of equal intensity and different +pitch. Riemann showed, by considerations of this kind, that the +essential foundation 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.[9] Then he showed that +the kind of free mobility without change of form which belongs to +bodies in our space can only exist when certain quantities yielded by +the calculation[10]--quantities that coincide with Gauss’s measure of +surface-curvature when they are expressed for surfaces--have everywhere +an equal value. For this reason Riemann 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,[11] I will once more observe that this so-called +measure of space-curvature is a quantity obtained by purely analytical +calculation, and that its introduction involves no suggestion of +relations that would have a meaning only for sense-perception. The +name is merely taken, as a short expression for a complex relation, +from the one case in which the quantity designated admits of sensible +representation. + +[Footnote 8: Ibid. p. 86.] + +[Footnote 9: 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.] + +[Footnote 10: They are algebraical expressions compounded from the +coefficients of the various terms in the expression for the square +of the distance of two contiguous points and from their differential +quotients.] + +[Footnote 11: As occurs, for instance, in the above-mentioned work of +Tobias, pp. 70, etc.] + +Now whenever the value of this measure of curvature in any space +is everywhere zero, that space everywhere 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 _pseudospherical_ 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 intersect another given straightest line in that surface. + +Beltrami[12] has rendered these last relations imaginable by showing +that the points, lines, and surfaces of a pseudospherical space of +three dimensions, can be so portrayed in the interior of a sphere +in Euclid’s homaloid 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 direction 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. + +[Footnote 12: _Teoria fondamentale, &c._, _ut sup._] + +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 +distinguishable from, zero. This latter definition is given in the +axioms of straight lines and parallels. + +Whilst Riemann entered upon this new field from the side of the most +general and fundamental questions of analytical geometry, I myself +arrived at similar conclusions,[13] 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. +Riemann starts by assuming the above-mentioned algebraical expression +which represents in the most general form the distance between two +infinitely near points, and deduces therefrom, 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. + +[Footnote 13: Ueber die Thatsachen die der Geometrie zum Grunde liegen +(_Nachrichten von der königl. Ges. d. Wiss. zu Göttingen_, Juni 3, +1868).] + +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 dimensions 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 definition can, in the first +instance, run only as follows: Between the co-ordinates of any two +points belonging to a solid body, there must be an equation which, +however 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 definition 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, + +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-ordinates 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 +dimensions 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 assumption for the definition of solidity, +becomes quite sufficient to determine the kind of equations holding +between 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 something that need not be. When in our space of three +dimensions two points of a solid body are kept fixed, its movements +are limited to rotations round the straight line connecting them. +If we turn it completely round once, it again occupies exactly the +position it had at first. This fact, that rotation in one direction +always brings a solid body back into its original 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 increased +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 +distinction of different spaces according to their measure of curvature. + +It still remained to be seen whether the laws of motion, as dependent +on moving forces, could also be consistently transferred to spherical +or pseudospherical space. This investigation has been carried out by +Professor Lipschitz of Bonn.[14] 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. + +[Footnote 14: ‘Untersuchungen über die ganzen homogenen Functionen von +_n_ Differentialen’ (Borchardt’s _Journal für Mathematik_, Bd. lxx. 3, +71; lxxiii. 3,1); ‘Untersuchung eines Problems der Variationsrechnung’ +(_Ibid._ Bd. lxxiv.).] + +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 curvature from that of Euclid’s flat +space. But if we can imagine such spaces of other sorts, it cannot be +maintained that the axioms of geometry are necessary consequences of an +_à priori_ transcendental form of intuition, 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 Riemann, 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 pseudospherical, 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 +immeasurably distant fixed stars equal to zero (in pseudospherical +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 Riemann 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. + +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 theodolite 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 geometrical 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 temperature, 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 measuring arcs and angles in the visual field. + +Every comparative estimate of magnitudes or measurement of their +spatial relations proceeds therefore upon a supposition as to the +behaviour of certain physical things, either the human body or other +instruments 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 increased 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 irregularities. A well-made convex mirror of moderate +aperture represents the objects in front of it as apparently 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 surface +of the mirror are found the images of all the other objects before +it, but the images are diminished and flattened in proportion to the +distance of their objects from the mirror. The flattening, or decrease +in the third dimension, is relatively greater than the decrease of +the surface-dimensions. Yet every straight line or every plane in the +outer world is represented by a straight line or a plane in the image. +The image of a man measuring with a rule a straight line from the +mirror would contract more and more the farther he went, but with his +shrunken rule the man in the image would count out exactly the same +number of centimetres as the real man. And, in general, all geometrical +measurements of lines or angles made with regularly varying images of +real instruments would yield exactly the same results as in the outer +world, all congruent bodies would coincide on being applied to one +another in the mirror as in the outer world, all lines of sight in the +outer world would be represented by straight lines of sight in the +mirror. In short I do not see how men in the mirror are to discover +that their bodies are not rigid solids and their experiences good +examples of the correctness of Euclid’s axioms. But if they could look +out upon our world as we can look into theirs, without overstepping +the boundary, they must declare it to be a picture in a spherical +mirror, and would speak of us just as we speak of them; and if two +inhabitants of the different worlds could communicate with one another, +neither, so far as I can see, would be able to convince the other that +he had the true, the other the distorted, relations. Indeed I cannot +see that such a question would have any meaning at all, so long as +mechanical considerations are not mixed up with it. + +Now Beltrami’s representation of pseudospherical space in a sphere of +Euclid’s space, is quite similar, except 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 mathematical expression.[15] 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. + +[Footnote 15: Compare the Appendix at the end of this Lecture.] + +We can even go a step further, and infer how the objects in a +pseudospherical world, were it possible to enter one, would appear +to an observer, whose eye-measure and experiences of space had been +gained like ours in Euclid’s space. Such an observer would continue +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,[16] 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 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. + +[Footnote 16: The reciprocal of the square of this distance, expressed +in negative quantity, would be the measure of curvature of the +pseudospherical space.] + +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 +prismatically to resemble pieces of one continuous larger lens. With +these, like the convex mirror, we see remote objects 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 +hundred 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 pseudospherical space. In short, pseudospherical +space would not seem to us very strange, comparatively speaking; we +should only at first be subject to illusions 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 prismatic +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 perspective picture. + +At the same time it must be noted that as a small elastic flat disk, +say of india-rubber, can only be fitted to a slightly curved spherical +surface with relative contraction 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 coherence being of course +maintained only in as far as their elasticity permitted their bending +without breaking. 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 sensible 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 represent to ourselves the look of +a pseudospherical world in all directions just as we can develop the +conception of it. Therefore it cannot be allowed that the axioms of our +geometry depend on the native form of our perceptive faculty, or are +in any way connected with it. + +It is different with the three dimensions of space. As all our means of +sense-perception extend only to space of three dimensions, and a fourth +is not merely a modification of what we have, but something perfectly +new, we find ourselves by reason of our bodily organisation quite +unable to represent a fourth dimension. + +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 compare, divide, and +measure them. All space-measurements, and therefore in general all +ideas of quantities applied to space, assume the possibility of figures +moving 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, however, +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 distensions 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 +velocity in a straight line, is not adapted to the image of the world +in the convex mirror. The path would indeed 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 than natural bodies do ever in fact correspond +exactly to the abstract notion we have obtained of them by induction. +Taking the notion of rigidity thus as a mere ideal, a strict Kantian +might certainly look upon the geometrical axioms as propositions given, +_à 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 +qualities and deportment a body must have to be recognised as rigid. + +But if to the geometrical axioms we add propositions relating to +the mechanical properties of natural bodies, were it only the axiom +of inertia, or the single proposition, that the mechanical and +physical properties 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 +experience. 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 elastic solids of great resistance, retain the same +form in every part of space if no external force affects them, is a +single case falling under the general principle. + +In conclusion, I do not, of course, maintain that mankind 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 impossible, of every geometrical +representation at variance with this fact. For this no knowledge +of the necessary 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 numerous and attentive observations--an intuition of +the kind the artist possesses of the objects he is to represent, and +by means of which he decides with certainty and accuracy whether a new +combination, which he tries, will correspond or not with their nature. +It is true that we have no word but _intuition_ to mark this; but it +is knowledge empirically gained by the aggregation and reinforcement +of similar recurrent impressions in memory, and not a transcendental +form given before experience. That other such empirical intuitions of +fixed typical relations, when not clearly comprehended, have frequently +enough been taken by metaphysicians for _à 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, represent no relations of real things. +When thus isolated, 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 geometry. + +(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 +observations, 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. + + +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 + + _x_² + _y_² +_z_² + _t_² = _R_² (1.) + +and for the distance _ds_ between the points (_x_, _y_, _z_, _t_) and +[(_x_ + _dx_) (_y_ + _dy_) (_z_ + _dz_) (_t_ + _dt_)] the value + + _ds_² = _dx_² + _dy_² + _dz_² + _dt_² (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 + + _ax_ + _by_ + _cz_ + _ft_ = 0 } + } (3.) + _αx_ + _βy_ + _γz_ + _φt_ = 0 } + +in which _a_, _b_, _c_, _f_, as well as _α_, _β_, _γ_, _φ_, are +constants. + +The length of the shortest arc, _s_, between the points + +(_x_, _y_, _z_, _t_), and (_ξ_, _η_, _ζ_, _τ_) follows, as in the +sphere, from the equation + + cos_s_ (_xξ_ + _yη_ + _zζ_ + _tτ_) + ------ = --------------------------- (4.) + _R_ _R_² + +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 + + _ξ_ = _η_ = _ζ_ = 0 + +from which equation 1 gives _τ_ = _R_, then, + + ( _s_₀ ) _σ_ + sin ( ---- ) = ----- + ( _R_ ) _R_ + +in which + + ____________________ + _σ_ = √(_x_² + _y_² + _z_²) + +or, + + ( _σ_ ) ( _σ_ ) + _s_₀ = _R_ . arc sin( --- ) = _R_ . arc tang( --- ) (5.) + ( _R_ ) ( _t_ ) + +In this, _s_₀ 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_ ) ( _Rz_ ) + χ = ( ---- ) ϒ = ( ---- ) ζ = ( ---- ) + ( _t_ ) ( _t_ ) ( _t_ ) + + _R_²_σ_² + χ² + ϒ² + ζ² = _r_² = --------- + _t_² + +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 χ, ϒ, ζ, by straight lines. For very small values of + +_x_, _y_, _z_, _t_ = _R_, and χ = _x_, ϒ = _y_, ζ = _z_ + +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 + + ( _r_ ) + _s_₀ = _R_ . arc tang( ± ---- ) (6.) + ( _R_ ) + +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_₀ < +½_R_π, one for which _s_₀ > ½_R_π. The extension in the direction of +_r_ is then + + _ds_₀ _R_² + ----- = ------------- + _dr_ _R_² + _r_² + +In order to obtain corresponding expressions for pseudospherical space, +let _R_ and _t_ be imaginary; that is, _R_ = ℛ_i_, and _t_ = τ_i_. +Equation 6 gives then + + _s_₀ _r_ + tang ------ = ± ------ + _i_ℛ _i_ℛ + +from which, eliminating the imaginary form, we get + + ℛ + _r_ + _s_₀ = ½ℛ log. nat. --------- + ℛ - _r_ + +Here _s_₀ has real values only as long as _r_ = R; for _r_ = ℛ the +distance _s_₀ 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 + + _ds_₀ ℛ² + ----- = ----------- + _dr_ ℛ² - _r_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 + + _____________________ + √_dx_² + _dy_² + _dz_² _t_ τ _σ_ + ---------------------- = ----- = ---- = ----- + _____________________ _R_ ℛ _r_ + √_d_χ² + _d_ϒ² + _d_ζ² + + __________________ + √_x_² + _y_² + _z_² + = ------------------- + _____________ + √χ² + ϒ² + ζ² + + + + +ON THE RELATION 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 address 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 frequent 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 physiology +of the senses; and in reference to those who have a long acquaintance +with, and who are quite at home in the beautiful fields of art, I may +compare myself to a traveller who has entered upon them by a steep and +stony mountain path, but who, in doing so, has passed many a stage from +which a good point of view is obtained. If therefore I relate to you +what I consider I have observed, it is with the understanding 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 condition 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 elementary 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 insight into its action, and into the principle of +its methods, cannot be complete if this physiological element is not +taken into account. Next to music this seems to predominate more +particularly in painting, and this is the reason why I have chosen +painting as the subject of my present lecture. + +The more immediate object of the painter is to produce in us by +his palette a lively visual impression of the objects which he has +endeavoured to represent. The aim, in a certain sense, is to produce a +kind of optical illusion; not indeed that, like the birds who pecked +at the painted grapes of Apelles, we are to suppose we have present +the real objects themselves, and not a picture; but in so far that the +artistic representation 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 instructive 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 accurate, and whose +memory for these images is particularly true. That which long tradition +has handed down to the men most gifted in this respect, and that +which they have found by innumerable experiments in the most varied +directions, as regards means and methods of representation, forms a +series of important and significant facts, which the physiologist, who +has here to learn from the artist, cannot afford to neglect. 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 determining 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 examination of the action of the +elementary means with which it works. The knowledge of the latter must, +however, form an indispensable basis for the solution of the deeper +questions, if we are to understand the problems which the artist has to +solve, and the mode in which he attempts to attain his object. + +I need scarcely lay stress on the fact, following as it does from +what I have already said, that it is not my intention to furnish +instructions according to which the artist is to work. I consider it a +mistake to suppose that any kind of æsthetic 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. + + +I. FORM. + +The painter seeks to produce in his picture an image of external +objects. The first aim of our investigation must be to ascertain what +degree and what kind of similarity he can expect to attain, and what +limits are assigned to him by the nature of his method. The uneducated +observer usually requires nothing more than an illusive resemblance +to nature: the more this is obtained, the more does he delight in the +picture. An observer, on the contrary, whose taste in works of art has +been more finely educated, will, consciously or unconsciously, require +something more, and something different. A faithful copy of crude +Nature he will at most regard as an artistic feat. To satisfy him, he +will need artistic selection, grouping, and even idealisation 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 +expression, and a characteristic development, and if possible beautiful +forms, which have perhaps belonged to no living individuals or indeed +any individuals which ever have existed, but only to such a one as +might exist, and as must exist, to produce a vivid perception of any +particular aspect of human existence in its complete and unhindered +development. + +If however the artist is to produce an artistic arrangement of only +idealised types, whether of man or of natural objects, must not the +picture be an actual, complete, and directly true delineation of that +which would appear if it anywhere came into being? + +Since the picture is on a plane surface, this faithful 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 photographer, gives on the +retina, which is its sensitive plate, only perspective views of the +external world; these are stationary, like the drawing on a picture, as +long as the standpoint of the eye is not altered. And, in fact, if we +restrict ourselves in the first place to the form of the object viewed, +and disregard for the present any consideration of colour, by a correct +perspective drawing we can present to the eye of an observer, who views +it from a correctly chosen point of view, the same forms of the visual +image as the inspection of the objects themselves would present to the +same eye, when viewed from the corresponding point of view. + +But apart from the fact that any movement of the observer, whereby his +eye changes its position, will produce displacements of the visual +image, different when he stands before objects from those when he +stands before the image, I could speak of only _one_ eye for which +equality of impression is to be established. 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 important 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 different perspective images. With the right eye +we see somewhat more of the right side of objects before us, and also +somewhat more of those behind it, than we do with the left eye; and +conversely we see with the left, more of the left side of an object, +and of the background 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 corresponding 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 different +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 displaced 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 binocular vision and on the movement +of the body are less operative, because in very distant objects the +differences between the images of the two eyes, or between 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 +powerfully 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 projections, 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, selection, and mode of illumination +of the objects represented. 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 +expression, 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 perspective. Nearer objects partially +conceal more distant ones, but can never themselves be concealed by +the latter. If therefore the painter skilfully groups his objects, 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 counteracts 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 perspective 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 horizontal 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 represent. + +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 foreground, and thus their apparent +magnitude furnishes a measure of the distance at which they are placed. + +Shadows, and more especially double ones, are of great importance. You +all know how much more distinct is the impression which a well-shaded +drawing gives as distinguished from an outline; the shading is hence +one of the most difficult, but at the same time most effective, +elements in the productions of the draughtsman and painter. It is his +task to imitate the fine gradation and transitions of light and shade +on rounded surfaces, which are his chief means of expressing 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, produce the impression of reliefs which are +only illuminated 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 completely without exception, that even in stereoscopic +views a falsely placed double shadow may destroy or confuse the entire +illusion. + +The various kinds of illumination are not all equally favourable for +obtaining the full effect of shadows. When the observer looks at the +objects in the same direction as that in which light falls upon them, +he sees only their illuminated sides and nothing of the shadow; the +whole relief which the shadows could give then disappears. If the +object is between the source of light and the observer he only sees the +shadows. Hence we need lateral illumination for a picturesque shading; +and over surfaces which like those of plane or hilly land only present +slightly moving figures, we require light which is almost in the +direction of the surface itself, for only such a one gives shadows. +This is one of the reasons which makes illumination by the rising or +the setting sun so effective. The forms of the landscape become more +distinct. To this must also be added the influence of colour, and of +aerial light, which we shall subsequently discuss. + +Direct illumination from the sun, or from a flame, makes the shadows +sharply defined, and hard. Illumination 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 +significance, is what is called _aerial perspective_. By this we +understand the optical action of the light, which the illuminated +masses of air, between the observer and distant objects, give. This +arises from a fine opacity in the atmosphere, which never entirely +disappears. If, in a transparent medium, there are fine transparent +particles of varying density and varying refrangibility, in so far as +they are struck by it, they deflect the light passing through such +a medium, partly by reflection and partly by refraction; to use an +optical expression, 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 distributed +in the transparent medium as an opaque halo. Water rendered turbid by +a few drops of milk shows this dispersion of the light and cloudiness +very distinctly. The light in this case is deflected by the microscopic +globules of butter which are suspended in the milk. + +In the ordinary air of our rooms, this turbidity is very apparent when +the room is closed, and a ray of sunlight is admitted through a narrow +aperture. We see then some of these solar particles, large enough to be +distinguished by the naked eye, while others form a fine homogeneous +turbidity. But even the latter must consist mainly of suspended +particles of organic substances, for, according to an observation of +Tyndall, they can be burnt. If the flame of a spirit lamp is placed +directly below the path of these rays, the air rising from the flame +stands out quite dark in the surrounding bright turbidity; that is to +say, the air rising from the flame has been quite freed from dust. +In the open air, besides dust and occasional smoke, we must often +also take into account the turbidity arising from incipient aqueous +deposits, where the temperature of moist air sinks so far that the +water retained in it can no longer exist as invisible vapour. Part of +the water settles then in the form of fine drops, as a kind of the very +finest aqueous dust, and forms a finer or denser fog; that is to say, +cloud. The turbidity which forms in hot sunshine and dry air may arise, +partly from dust which the ascending currents of warm air whirl about; +and partly from the irregular mixture of cold and warm layers of air +of different density, as is seen in the tremulous motion of the lower +layers of air over surfaces irradiated by the sun. But science can as +yet give no explanation of the turbidity in the higher regions of the +atmosphere which produces the blue of the sky; we do not know whether +it arises from suspended particles of foreign substances, or whether +the molecules of air themselves may not act as turbid particles in the +luminous ether. + +The colour of the light reflected by the opaque particles mainly +depends on their magnitude. When a block of wood floats on water, and +by a succession of falling drops we produce small wave-rings near +it, these are repelled by the floating wood as if it were a solid +wall. But in the long waves of the sea, a block of wood would be +rocked about without the waves being thereby materially disturbed in +their progress. Now light is well known to be an undulatory motion +of the ether which fills all space. The red and yellow rays have the +longest waves, the blue and violet the shortest. Very fine particles, +therefore, which disturb the uniformity of the ether, will accordingly +reflect the latter rays more markedly than the red and yellow rays. The +light of turbid media is bluer, the finer are the opaque particles; +while the larger particles of uniform light reflect all colours, and +therefore give a whitish turbidity. Of this kind is the celestial +blue, that is, the colour of the turbid atmosphere as seen against +dark cosmical space. The purer and the more transparent the air, the +bluer is the sky. In like manner it is bluer and darker when we ascend +high mountains, partly because the air at great heights is freer +from turbidity, and partly because there is less air above us. But +the same blue, which is seen against the dark celestial space, also +occurs against dark terrestrial objects; for instance, when a thick +layer of illuminated air is between us and masses of deeply shaded or +wooded hills. The same aerial light makes the sky blue, as well as the +mountains; excepting that in the former case it is pure, while in the +latter it is mixed with the light from objects behind; and moreover +it belongs to the coarser turbidity of the lower regions of the +atmosphere, so that it is whiter. In hot countries, 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 yellowish 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, especially snow-mountains, appear coloured. + +These colourations are moreover not peculiar to the air, but occur +in all cases in which a transparent substance is made turbid by the +admixture of another transparent substance. We see it, as we have +observed, 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, separate 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. Goethe called +attention to the universality of this phenomenon, and endeavoured 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 mountainous 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 magnitudes 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. + + +II. SHADE. + +The circumstances which we have hitherto discussed indicate a profound +difference, and one which is exceedingly important for the perception +of solid form, between the visual image which our eyes give, when we +stand before objects, and that which the picture gives. The choice +of the objects to be represented in pictures is thereby at once much +restricted. Artists are well aware that there is much which cannot be +represented by the means at their disposal. Part of their artistic +skill consists in the fact that by a suitable grouping, position, +and turn of the objects, by a suitable choice of the point of view, +and by the mode of lighting, they learn to overcome the unfavourable +conditions which are imposed on them in this respect. + +It might at first sight appear that of the requisite truth to nature +of a picture, so much would remain that, seen from the proper point +of view, it would at least produce the same distribution of light, +colour, and shadow in its field of view, and would produce in the +interior of the eye exactly the same image on the retina as the object +represented would do if we had it actually before us, and looked at it +from a definite, fixed point of view. It might seem to be an object of +pictorial skill to aim at producing, under the given limitations, the +_same_ effect as is produced by the object itself. + +If we proceed to examine whether, and how far, painting can satisfy +such a condition, we come upon difficulties before which we should +perhaps shrink, if we did not know that they had been already overcome. + +Let us begin with the simplest case; with the quantitative 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 picture-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 admixtures; 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 intensities 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 Lambert’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 diameter. Its density and illuminating power is here only +the one forty-thousandth of that with which it left the sun’s surface; +and Lambert’s number leads to the conclusion that even the brightest +white surface on which the sun’s rays fall vertically, has only the one +hundred-thousandth part of the brightness of the sun’s disk. The moon +however is a gray body, whose mean brightness is only about one fifth +of that of the purest white. + +And when the moon irradiates a body of the purest white on the earth, +its brightness is only the hundred-thousandth part of the brightness of +the moon itself; hence the sun’s disk is 80,000 million times brighter +than a white which is irradiated by the full moon. + +Now pictures which hang in a room are not lighted by the direct light +of the sun, but by that which is reflected 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 1-20th of the brightness of +white directly lighted by the sun; it will generally be only 1-40th, or +even less. + +Hence the painter of the desert, even if he gives up the representation +of the sun’s disk, which is always very imperfect, will have to +represent the glaringly lighted garments of his Bedouins with a white +which, in the most favourable case, shows only the 1-20th 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-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 represent 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 ¹/₁₀₀ that of white paper. The brightest colours of a +painter are only about one hundred times as bright as his darkest +shades. + +The statements I have made may perhaps appear exaggerated. But they +depend upon measurements, and you can control them by well-known +observations. According to Wollaston, the light of the full moon is +equal to that of a candle burning at a distance of 12 feet. You know +that we cannot read by the light of the full moon, though we can read +at a distance of three or four feet from a candle. Now assume that you +suddenly passed from a room in daylight to a vault perfectly dark, with +the exception of the light of a single candle. You would at first think +you were in absolute darkness, and at most you would only recognise the +candle itself. In any case, you would not recognise the slightest trace +of any objects at a distance of 12 feet from the candle. These however +are the objects whose illumination is the same as that which the +moonlight gives. You would only become accustomed to the darkness after +some time, and you would then find your way about without difficulty. + +If, now, you return to the daylight, which before was perfectly +comfortable, it will appear so dazzling that you will perhaps have +to close the eyes, and only be able to gaze round with a painful +glare. You see thus that we are concerned here not with minute, but +with colossal, differences. How now is it possible that, under such +circumstances, we can imagine there is any similarity between the +picture and reality? + +Our discussion of what we did not see at first, but could afterwards +see in the vault, points to the most important element in the solution; +it is the varying extent to which our senses are deadened by light; +a 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 insensitive 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 moonlight 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 certain 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 transcript of the object, but a translation of his impression +into another scale of sensitiveness, which belongs to a different +degree of impressibility of the observing eye, in which the organ +speaks a very different dialect in responding to the impressions of the +outer world. + +In order to understand to what conclusions this leads, I must first +of all explain the law which Fechner discovered for the scale of +sensitiveness of the eye, which is a particular case of the more +general _psycho-physical_ law of the relations of the various sensuous +impressions to the irritations which produce them. This law may +be expressed as follows: _Within very wide limits of brightness, +differences in the strength of light are equally distinct or appear +equal in sensation, if they form an equal fraction of the total +quantity of light compared._ Thus, for instance, differences in +intensity of one hundredth of the total amount can be recognised +without great trouble with very different strengths of light, without +exhibiting material differences in the certainty and facility of the +estimate, whether the brightest daylight or the light of a good candle +be used. + +[Illustration: FIG. 3.] + +[Illustration: FIG. 4.] + +The easiest method of producing accurately measurable differences in +the brightness of two white surfaces, depends on the use of rapidly +rotating disks. If a disk, like the adjacent one in Fig. 3, is made to +rotate very rapidly (that is, 20 to 30 times in a second), it appears +to the eye to be covered with three grey rings as in Fig. 4. The +reader must, however, figure to himself the grey of these rings, as +it appears on 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 compared with the length of +half the circumference of the corresponding ring, we get the fraction +by which the intensity of the light in the white ground of the disk is +diminished in the ring in question. If the bands are all equally broad, +as in Fig. 3, the inner rings appear darker than the outer ones, for in +this latter case the same loss of light is distributed over a larger +area than in the former. In this way extremely delicate shades of +brightness may be obtained, and by this method, when the strength of +the illumination varies, the brightness always diminishes by the _same +proportion_ of its total value. Now it is found, in accordance with +Fechner’s law, that the distinctness of the rings is nearly constant +for very different strengths of light. We exclude, 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 shadow, +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 comparison with the newly added daylight, provided that this strikes +all parts of the white sheet uniformly. You see then that, while the +difference between candle-light and darkness can be easily perceived, +the equally great difference between daylight, on the one hand, and +daylight 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 reflects a small fraction, of incident light. For different +intensities of illumination, the difference of brightness between +the two will always correspond to the same fraction 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, notwithstanding 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 absolute 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 brightness is for us the perception, from +which we form our 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 nature, to evoke in us the same +conception as to the nature of the bodies seen. A truthful imitation in +this respect would be attained within the limits in which Fechner’s law +holds, if the artist reproduced the fully lighted parts of the objects +which he has to represent with pigments, which, with the same light, +were equal to the colours to be represented. This is approximately the +case. On the whole, the painter chooses coloured pigments which almost +exactly reproduce the colours of the bodies represented, especially +for objects of no great depth, such as portraits, and which are only +darker in the shaded parts. Children begin to paint on this principle; +they imitate one colour by another; and, in like manner also, nations +in which painting has remained in a childish stage. Perfect artistic +painting is only reached when we have succeeded in imitating the action +of light upon the eye, and not merely the pigments; and only when we +look at the object of pictorial representation from this point of view, +will it be possible to understand the variations from nature which +artists have to make in the choice of their scale of colour and of +shade. + +These are, in the first case, due to the circumstance that Fechner’s +law only holds for mean degrees of brightness; while, for a brightness +which is too high or too low, appreciable divergences are met with. + +At both extremes of luminous intensity the eye is less sensitive for +differences in light than is required by that law. With a very strong +light it is dazzled; that is, its internal activity cannot keep pace +with the external excitations; the nerves are too soon tired. Very +bright objects appear almost always to be equally bright, even when +there are, in fact, material differences 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 illumination, darker objects +become more like the darkest objects, while with greater illumination +brighter objects become more like the brightest than should be the +case in accordance with Fechner’s law, which holds for mean degrees of +illumination. From this results, what, for painting, is an extremely +characteristic difference between the impression of very powerful and +very feeble illumination. + +When painters wish to represent glowing sunshine, they make all objects +almost equally bright, and thus produce with their moderately bright +colours the impression which the sun’s glow makes upon the dazzled +eye of the observer. If, on the contrary, they wish to represent +moonshine, they only indicate the very brightest objects, particularly +the reflection of moonlight 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 brightness 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 impression on the eye as is produced by equal degrees +of brightness of actual objects. The alteration in the scale of +shade which has been described is necessary because the colours of +the picture are seen in the mean brightness of a moderately lighted +room, for which 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 corresponds 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 aërial 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 remarkable in this shading, and yet these pictures +give particularly bright and vivid aspects of the objects. Hence +they are of particular interest for understanding the principles of +pictorial illumination. + +In order to explain these actions we must, I think, consider that +while Fechner’s law is approximately correct for those mean lights +which are agreeable to the eye, the deviations which are so marked, +for too high or too 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 produced 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 ¹/₁₅₀ or ¹/₁₈₀ of the total intensity can be recognised. The light +in which pictures are looked at is, on the contrary, much feebler; and +if we are to retain the same distinctness of the finest shadows and +of the modelling of the contours which it produces, the gradations of +shade in the picture must be somewhat stronger than corresponds to the +exact luminous intensities. The darkest objects of the picture thereby +become unnaturally 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 gradations are imitated with expression. + + +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 admixtures of +three simple sensations; namely, of red, green, and violet,[17] which, +by a not improbable supposition of Thomas Young, can be apprehended +quite independently of each other by three different systems of +nerve-fibres. To this independence of the different sensations of +colour corresponds their independence in the gradation of intensity. +Recent measurements[18] have shown that the sensitiveness of our eye +for feeble shadows is greatest in the blue and least in the red. A +difference of ¹/₂₀₅ to ¹/₂₆₈ of the intensity can be observed in the +blue, and with an untired eye of ¹/₁₆ in the red; or when the colour is +dimmed by being looked at for a long time, a difference of ¹/₅₀ to ¹/₇₀. + +[Footnote 17: Helmholtz’s _Popular Scientific Lectures_, pp. 232-52.] + +[Footnote 18: Dobrowolsky in _Graefe’s Archiv für Ophthalmologie_, vol. +xviii. part i. pp. 24-92.] + +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, 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 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 difficult, 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 imitating, 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 moonlight, 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 grey. + +I will again remind you that these changes of 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, previously 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 understand 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, _Chevreul’s simultaneous 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, however, enter upon them, +for they produce no divergence 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 impression of each colour changes, +for it is depicted on portions 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 movements of the eye, and has been called by +Chevreul, ‘_successive Contrast_.’ + +We have already seen that the retina is more sensitive 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 perceived. This latter process is designated as ‘Fatigue’ of +the retina; an exhaustion of the capability of the retina by its own +activity, just as the muscles by their activity become tired. + +I must here remark that the fatigue of the retina by light does not +necessarily extend to the whole surface; but when only a small portion +of this membrane is struck by a minute, defined picture it can also be +locally developed in this part only. + +You must all have observed the dark spots which move about in the field +of vision, when we have been looking for only a short time towards the +setting sun, and which physiologists call _negative after-images_ of +the sun. They are due to the fact that only those parts of the retina +which are actually struck by the image of the sun in the eye, have +become insensitive to a new impression of light. If, with an eye which +is thus locally tired, we look towards a uniformly bright surface, 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 +impressions of light. A longer time is required in order to develop +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 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.[19] 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,[20] 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. 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. + +[Footnote 19: 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.] + +[Footnote 20: See Helmholtz’s _Popular Lectures_, first series, p. 250.] + +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 shifting about on the retina, +has moreover the advantage of avoiding disturbances of sight, which +powerful and continuous after-images would bring with them. Yet here +also, after-images are not wanting; only they are shadowy in their +contours, and of very short duration. + +If a red surface be laid upon a grey ground, and if we look from the +red over the edge towards the grey, the edges of the grey will seem as +if struck by such an after-image of red, and will seem to be of a faint +bluish green. But as the after-image rapidly disappears, it is mostly +only those parts of the grey, which are nearest the red, which show the +change in a marked degree. + +This also is a phenomenon which is produced more strongly by bright +light and brilliant saturated colours than by fainter light and +duller colours. The artist however, works for the most part with the +latter. He produces most of his tints by mixture; each mixed pigment +is, however, greyer and duller than the pure colour of which it is +mixed, and even the few pigments 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 contrasts, 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 +themselves 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 background of blue, and a rose-red tint +where they impinge 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 predominant +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 represented by the painter, since the colours of his pictures +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 bright 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 +neighbourhood 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 disappears, which covers the whole neighbourhood, and, at the same +time, any objects there may be in the dark part of the field of view +are seen more distinctly. If the flame is partly screened by a ruler, +this appears jagged where the flame projects beyond it. The luminosity +in the neighbourhood of the flame is so intense, 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 aërial +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 aërial 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 perfectly 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 crystal. 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.[21] + +[Footnote 21: I disregard here the view that irradiation in the eye +depends on 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.] + +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 phenomena 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 have given, or would give, +which is to be imitated, so as to produce as distinct and vivid a +conception as possible 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 +canvas. The fact that they are represented is particularly 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 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 pictorial +elements. + + +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 imitating 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 obviating these +evils? Such there are indeed. Frescoes are sometimes viewed in direct +sunlight; transparencies 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 pictures which +took advantage of this. Fresco painting would have led to this; or the +experiments of Münich’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 moderation of light and of +colours in pictures is ever advantageous, and we need only look at +frescoes in direct sunlight, such as those of the new Pinakothek in +Münich, 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, brilliant +colours were used, even locally and to a moderate extent, which were +intended to represent bright sunlight, and a mass of light shed over +the picture. It is much easier to produce an accurate imitation of the +feeble light of moonshine with artificial light in dioramas and theatre +decorations. + +We may therefore designate truth to Nature of a beautiful picture as +an ennobled fidelity to Nature. Such a picture reproduces all that is +essential in the impression, and attains full vividness of conception, +but without injury or tiring the eye by the nude lights of reality. The +differences between Art and Nature are chiefly confined, as we have +already seen, to those matters which we can in reality only estimate +in an uncertain 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 perceiving the outer world, and which admit of the finest +discrimination and observation. + +It has been mentioned above that the discrimination 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 +attention 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 appears 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 construction of living +organisms, whatever their origin, excludes the possibility that in the +majority of healthy individuals an instinct should be developed or +maintain itself which did not serve some definite purpose. + +We have not far to seek for the delight in light and in colours, and +for the dread of darkness; this coincides with the endeavour to see +and to recognise surrounding objects. Darkness owes the greater part +of the terror which it inspires to the fright of what is unknown and +cannot be recognised. A coloured picture gives a far more accurate, +richer, and easier conception than a similarly executed drawing, which +only retains the contrasts of light and shade. A picture retains the +latter, but has in addition the material for discrimination which +colours afford; by which surfaces which appear equally bright in the +drawing, owing to their different colour, are now assigned to various +objects, or again as alike in colour are seen to be parts of the same, +or of similar objects. In utilising the relations thus naturally given, +the artist, by means of prominent colours, can direct and enchain the +attention of the observer upon the chief objects of the picture; and by +the variety of the garments he can discriminate the figures from each +other, but complete each individual one in itself. Even the natural +pleasure in pure, strongly saturated colours, finds its justification +in this direction. The case is analogous to that in music, with the +full, pure, well-sounding tones of a beautiful voice. Such a one is +more expressive; that is, even the smallest change of its pitch, or its +quality--any slight interruption, any tremulousness, any rising or +falling in it--is at once more distinctly recognised by the hearer than +could be the case with a less regular sound; and it seems also that +the powerful excitation which it produces in the ear of the listener, +arouses trains of ideas and passions more strongly than does a feebler +excitation 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 sensitive 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 perception 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 produces 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 +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 expressive 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, especially 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 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 one-sided 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 irritation 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 produces 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 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 appear of a carmine, or on a straw-colour they appear +of a golden-yellow. + +If we pass beyond these limits, we arrive at unpleasant 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 +combinations 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 combination tends towards yellow, and in the former, 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 one-sided fatigue of +the eye, without falling into the baldness of complementary 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 approximately +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 consonance of tones. On the +contrary, a consideration of the facts shows that a number of accessory +influences come into play,[22] when once the coloured surface is also +to produce, either wholly or in part, a representation of natural +objects or of solid forms, or even if it only offers a resemblance with +the representation of a relief, of shaded and of non-shaded surfaces. +It is moreover often difficult to establish, as a matter of fact, +what are the colours which produce the harmonic impression. This is +pre-eminently the case with pictures in which the aërial colour, the +coloured reflection 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 determination 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 representation, +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. + +[Footnote 22: Conf. E. Brücke, _Die Physiologie der Farben für +die Zwecke der Kunstgewerbe_. Leipzig, 1866. W. v. Bezold, _Die +Farbenlehre, ein Hinblick auf Kunst und Kunstgewerbe_. Braunschweig, +1874.] + +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 corresponds to the dominant light. This is occasioned, in the +first case, by the truthful imitation of physical circumstances. If +the illumination is rich in yellow light, yellow colours will appear +brighter and more brilliant than blue ones; for yellow bodies are +those which preferably reflect yellow light; while that of blue is +only feebly reflected, and is mainly absorbed. Before the shaded +parts of blue bodies, the yellow aërial 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 æsthetic requirements of artistic unity of +composition 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 aërial 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 distinct 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 binocular 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 position +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 aërial 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 transformed in the picture to one differing +sometimes by a hundredfold; how here, the colour of the object cannot +be simply represented by the pigment; that indeed it is necessary to +introduce important changes in the distribution of light and dark, of +yellowish and of bluish tints. + +The artist cannot transcribe Nature; he must translate her; yet this +translation may give us an impression in the highest degree distinct +and forcible, not merely of the objects themselves, but even of the +greatly altered intensities of light under which we view them. The +altered scale is indeed in many cases advantageous, as it gets rid +of everything which, in the actual objects, is too dazzling, and too +fatiguing for the eye. Thus the imitation of Nature in the picture is +at the same time an ennobling of the impression 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 impressions, and the feeling of delight associated +therewith, to direct and enchain the attention; it can use their +variety to heighten the direct understanding of what is represented, +and yet keep the eye in a condition of excitation most favourable and +agreeable for delicate sensuous impressions. + +If, in these considerations, my having continually laid much weight on +the lightest, finest, and most accurate sensuous intelligibility of +artistic representation, may seem to many of you as a very subordinate +point--a point which, if mentioned at all by writers on æsthetics, 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 +corresponding feelings, and direct them towards a common object, +so as to give a vivid perception of all the features 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 something 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 certainly, 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 comprehend 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. + + + + +ON THE ORIGIN OF THE PLANETARY 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, +hypotheses, or dreams. + +Of all the subjects to which the thought and imagination 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. +Beneficent and malignant deities, giants, Kronos who devours his +children, Niflheim, with the ice-giant Ymir, who is killed by the +celestial Asas,[23] that out of him the world may be constructed--these +are all figures which fill the cosmogonic systems of the more +cultivated 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. + +[Footnote 23: Cox’s _Aryan Mythology_, vol. i. 372. Longmans.] + +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 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 originating in some adventitious +circumstance; at the same time the matter they contain is comparatively +without originality, and they are only important from a destructive 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 philosophy, +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 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 energetically 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 conclusions 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, +however, receive direct confirmation, if, in the progress of scientific +knowledge, new facts accrue to those already known, and like them +are explained on the hypothesis; and particularly if survivals of the +processes, assumed to have taken place in the formation of the heavenly +bodies, can be proved to exist in the present. + +Such direct confirmations of various kinds have, in fact, been formed +for the view we are about to discuss, and have materially increased its +probability. + +Partly this fact, and partly the fact that the hypothesis in question +has recently been mentioned in popular and scientific books, in +connection with philosophical, ethical, and theological questions, have +emboldened me to speak of it here. I intend not so much to tell you +anything substantially new in reference 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 uniformity of +natural laws, our conclusions from present circumstances as to the +past, and as to the future, imperatively lead to an impossible state +of things; that is, to the necessity of an infraction of natural laws, +of a beginning which could not have been due to processes known to us. +Hence, to begin such an investigation as to the possible or probable +primeval history of our present world, is, considered as a question of +science, no idle speculation, but a question as to the limits of its +methods, and as to the extent to which existing laws are valid. + +It may perhaps appear rash that we, restricted as we are, in the circle +of our observations in space, by our position on this little earth, +which is but as a grain of dust in our milky way; and limited in time +by the short duration of the human race; that we should attempt to +apply the laws which we have deduced from the confined circle of facts +open to us, to the whole range of infinite space, and of time from +everlasting to everlasting. But all our thought and our action, in the +greatest as well as in the least, is based on our confidence in the +unchangeable order of nature, and this confidence has hitherto been the +more justified, the deeper we have penetrated into the interconnections +of natural phenomena. And that the general laws, which we have found, +also hold for the most distant vistas of space, has acquired strong +actual confirmation during the past half-century. + +In the front rank of all, then, is the law of gravitation. 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 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 +instance, 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 yourselves +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 eliminated, in this case, as they are excluded in +the planetary system. + +[Illustration: FIG. 5.] + +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 distances 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 be +circular but elliptical, of the form of the curved line in Fig. 5. But +these ellipses lie in both cases differently as regards the attracting +centre. In our model, the attracting force is stronger, the further +the lead sphere is removed from its position of equilibrium. Under +these circumstances, the ellipse of the path has such a position in +reference to the attracting centre, that this is in the centre, _c_, +of the ellipse. For planets, on the contrary, the attracting force +is feebler the 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 _b_, 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 publishing his important discovery +until it had acquired a direct confirmation; this followed from the +observations, 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 astronomical observation, increased +so far that all the complicated actions, which take place between +all the planets, and all their satellites, in consequence of +the mutual action of each upon each, and which astronomers 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 manifold phenomena which follow therefrom, +was so complete 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 attraction of our sun that +the law of gravitation was found to hold. With regard to the fixed +stars, it was found that double stars moved about each other in +elliptical paths, and that therefore the same law of gravitation must +hold for them as for our planetary system. The distance of some of +them could be calculated. The nearest of them, α, in the constellation +of the Centaur, is 270,000 times further from the sun than the earth. +Light, which has a velocity of 186,000 miles a second, which traverses +the distance from the sun to the earth in eight minutes, would take +four years to travel from α 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 can only be +seen by the most perfect instruments. The mass of Sirius is found to +be 13·76, and that of its satellite 6·71, times the mass of the sun; +their mutual distance is equal to thirty-seven times the radius of the +earth’s orbit, and is therefore somewhat larger than the distance of +Neptune from the sun. + +Another fixed star, Procyon, is in the same case as Sirius, but its +satellite has not yet been discovered. + +You thus see that in gravitation we have discovered 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 nebulæ. + +You all know that a fine bright line of light, seen through a glass +prism, appears as a coloured band, red and yellow at one edge, blue +and violet at the other, and green in the middle. Such a coloured +image is called a spectrum--the rainbow is such a one, produced by the +refraction of light, though not exactly by a prism; and it exhibits +therefore the series of colours into which white sunlight can thus be +decomposed. The formation of the prismatic spectrum depends on the +foot 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 substance vaporised by heat. The spectrum +of such a body consists, then, of one or more, and sometimes even a +great number, of entirely distinct bright lines, whose position and +arrangement in the spectrum is characteristic for the substances of +which the gas or vapour consists, so that it can be ascertained, by +means of spectrum analysis, what is the chemical constitution of the +ignited gaseous body. Gaseous spectra of this kind are shown in the +heavenly space by many nebulæ; 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 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 +nebulæ 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. + +[Illustration: FIG. 6.] + +[Illustration: FIG. 7.] + +[Illustration: FIG. 8.] + +Far more powerful apparatus was afterwards used by Kirchhoff, and then +by Angström, to push the decomposition of light as far as possible. +Fig. 6 represents 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, representing a fine slice of light, which +can be narrowed or widened by the small screw, and by which the +light under investigation can be allowed to enter. It then passes +through the telescope B, afterwards through the four prisms, and +finally through the telescope A, from which it reaches the eye of the +observer. Figs. 7, 8, and 9 represent small portions of the solar +spectrum as mapped by Kirchhoff, 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 +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. + +[Illustration: FIG. 9.] + +It follows from this, that the solar atmosphere contains 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, +moreover, 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 constituted; 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 α 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 of our +atmosphere, and hydrogen, an element in water, which indeed is formed +by its combustion. Both have been found in the irresolvable nebulæ, +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 terrestrial 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, surrounded 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 magnesium, 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.[24] These +structures could formerly only be viewed at the time of a total eclipse +of the sun, forming what were called the rose-red protuberances. 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. + +[Footnote 24: 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.] + +[Illustration: FIG. 10.] + +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 a spot +according to Padre Secchi, as seen under powerful 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 they are again resolved, +and meanwhile several rotations 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 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. + +[Illustration: FIG. 11.] + +Just on the edge of these spots there are spectroscopic 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 cooler gases from the outer +layers of the sun’s atmosphere 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 +expansion, and partly by radiation into space, must become cooled. +At the same time, they deposit their more difficultly volatile +constituents as fog or cloud. This cooling can only, of course, be +regarded as comparative; 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 atmosphere, 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 permanence 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 planetary disturbances. By disturbances of the +planetary 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 constant ellipse whose axes would retain the +same direction 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 disturbances due +to the reciprocal attraction of planets upon each other, and on the +assumption that no forces of other kinds have any influence on their +motions. + +On our earth we cannot produce such an everlasting 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, according 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 surrounding air in vibration, and +imparts to it some of its own motion. Thus it happens that its velocity +is continually 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 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, without 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 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 luminiferous ether.[25] + +[Footnote 25: This calculation would, however, lose its bases if +Maxwell’s hypothesis were confirmed, according to which light depends +on electrical and magnetical oscillations.] + +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 in the motion +of the planets since the time in which we possess astronomical +observations of their path. It is different with the smaller bodies +of our system. Encke in particular has shown, with reference to the +well-known small comet which bears his name, that it circulates round +the sun in ever-diminishing orbits and in ever shorter periods of +revolution. Its motion is similar to that of the circular pendulum +which we have mentioned, and which, having its velocity gradually +delayed by the resistance of the air, describes circles about its +centre of attraction, which continually become smaller and smaller. The +reason for this phenomenon is the following: The force which offers +a resistance to the attraction of the sun on all comets and planets, +and which prevents them from getting continually nearer to the sun, is +what is called the centrifugal force--that is, the tendency to continue +their motion in a straight line in the direction of their path. As +the force of their motion diminishes, they yield by a corresponding +amount to the attraction of the sun, and get nearer to it. If the +resistance continues, they will continue to get nearer the sun until +they fall into it. Encke’s comet is no doubt in this condition. But the +resistance whose presence in space is hereby indicated, must act, and +has long continued to act, in the same manner on the far larger masses +of the planets. + +The presence of partly fine and partly coarse heavy masses diffused +in cosmical space is more distinctly 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 atmosphere 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 terrestrial 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 shooting-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 atmosphere, 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 becoming 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 proportional to the work expended. If, like the +mechanicians, 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 sufficient +to raise the same weight of water through one degree Centigrade. The +equivalent in work of a velocity of eighteen to twenty-four miles in a +second may be easily calculated from known mechanical laws; and this, +transformed into heat, would be sufficient to raise the temperature of +a piece of meteoric iron to 900,000 to 2,500,000 degrees Centigrade, +provided that all the heat were retained by the iron, and did not, +as it undoubtedly does, mainly pass into the air. This calculation +shows, at any rate, that the velocity of the shooting-stars is +perfectly adequate to raise them to the most violent incandescence. The +temperatures attainable by terrestrial means scarcely exceed 2,000 +degrees. In fact, the outer crusts of meteoric stones generally show +traces of incipient fusion; and in cases in which observers examined +with sufficient promptitude 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 calculated 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 which move in regular elliptical orbits, cutting +the earth’s orbit in definite places, and therefore always occur on +particular days of the year. Thus the 10th of August of each year is +remarkable, and every thirty-three years the splendid fireworks of the +12th to the 14th of November repeats itself for a few years. It is +remarkable that certain comets accompany the paths of these swarms, and +give rise to the supposition that the comets gradually split up into +meteoric swarms. + +This is an important process. What the earth does is done by the other +planets, and in a far higher degree by the sun, towards which all the +smaller bodies of our system must fall; those, therefore, that are +more subject to the influence of the resisting medium, and which must +fall the more rapidly, the smaller they are. The earth and the planets +have for millions of years been sweeping together the loose masses in +space, and they hold fast what they have once attracted. But it follows +from this that the earth and the planets were once smaller than they +are now, and that more mass was diffused in space; and if we follow +out this consideration it takes us back to a state of things in which, +perhaps, all the mass now accumulated in the sun and in the planets, +wandered loosely diffused in space. If we consider, further, that the +small masses of meteorites as they now fall, have perhaps been formed +by the gradual aggregation of fine dust, we see ourselves led to a +primitive condition of fine nebulous masses. + +From this point of view, that the fall of shooting-stars and of +meteorites is perhaps only a small survival of a process which once +built up worlds, it assumes far greater significance. + +This would be a supposition of which we might admit the possibility, +but which could not perhaps claim any great degree of probability, if +we did not find that our predecessors, starting from quite different +considerations, had arrived at the same hypothesis. + +You know that a considerable number of planets rotate around the +sun besides the eight larger ones, Mercury, Venus, the Earth, Mars, +Jupiter, Saturn, Uranus, and Neptune; in the interval between Mars +and Jupiter there circulate, as far as we know, 156 small planets or +planetoids. Moons also rotate about the larger planets--that is, about +the Earth and the four most distant ones, Jupiter, Saturn, Uranus, and +Neptune; and lastly the Sun, and at any rate the larger planets, rotate +about their own axes. Now, in the first place, it is remarkable that +all the planes of rotation of the planets and of their satellites, +as well as the equatorial planes of these planets, do not vary much +from each other, and that in these planes all the rotation is in the +same direction. The only considerable exceptions known are the moons +of Uranus, 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, especially 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 +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 +primitive 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 considerations 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 extended 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 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 another 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 condensation of loosely diffused masses to +form larger bodies is by no means complete, but still goes on, though +the traces are slight. The form in which it now appears is altered by +the fact that meanwhile the gaseous or dust-like mass diffused in space +had united under the influence of the force of attraction, and of the +force of crystallisation of their constituents, to larger pieces than +originally existed. + +The showers of stars, as examples now taking place of the process which +formed the heavenly bodies, are important from another point of view. +They develop light and heat; and that directs us to a third series of +considerations, which leads again to the same goal. + +All life and all motion on our earth is, with few exceptions, kept up +by a single force, that of the sun’s rays, which bring to us light +and heat. They warm the air of the hot zones, this becomes lighter +and ascends, while the colder air flows towards the poles. Thus is +formed the great circulation of the passage-winds. Local differences +of temperature over land and sea, plains and mountains, disturb the +uniformity of this great motion, and produce for us the capricious +change of winds. Warm aqueous vapours ascend with the warm air, become +condensed into clouds, and fall in the cooler zones, and upon the +snowy tops of the mountains, as rain and as snow. The water collects +in brooks, in rivers, moistens the plains, and makes life possible; +crumbles the stones, carries their fragments along, and thus works at +the geological transformation of the earth’s surface. It is only under +the influence of the sun’s rays that the variegated covering of plants +of the earth grows; and while they grow, they accumulate 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 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 circulation 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 distantly compared. I have previously +explained the mechanical equivalent of heat. Calculated by that +standard, the work which the sun produces by its 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 independent of the special +interests of man. These have been generalised and comprehended in the +all-ruling natural law of the _Conservation of Force_. No natural +process, and no series of natural processes, can be found, however +manifold may be the changes which take place among them, by which a +motive force can be continuously produced without a corresponding +consumption. 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 matter itself. It seems as if Goethe +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 concerns us here: Whence +does the sun derive this enormous store of force which it sends out? + +On earth the processes of combustion are the most abundant source of +heat. Does the sun’s heat originate in a process of this kind? To this +question we can reply with a complete and decided negative, for we +now know that the sun contains the terrestrial elements with which we +are acquainted. Let us select from among them the two, which, for the +smallest mass, produce the greatest amount of heat when they combine; +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 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 extended to +millions of years. + +Known chemical forces are thus so completely inadequate, 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 +extraordinarily 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 +mountains works our mills. + +If a weight falls from a height and strikes the ground its mass loses, +indeed, the visible motion which it had as a whole--in fact, however, +this motion is not lost; it is transferred to the smallest elementary +particles of the mass, and this invisible vibration of the molecules is +the motion of heat. Visible motion is transformed by impact, into the +motion of heat. + +That which holds in this respect for gravity, holds also for +gravitation. A heavy mass, of whatever kind, which is suspended in +space separated from another heavy mass, represents a force capable +of work. For both masses attract each other, and, if unrestrained by +centrifugal force, they move towards each other under the influence of +this attraction; this takes place with ever-increasing velocity; and +if this velocity is finally destroyed, whether this be suddenly, by +collision, or gradually, by the friction of movable parts, it develops +the corresponding quantity of the motion of heat, the amount of which +can be calculated from the equivalence, previously established, between +heat and mechanical work. + +Now we may assume with great probability that very many more meteors +fall upon the sun than upon the earth, and with greater velocity, too, +and therefore give more heat. Yet the hypothesis, that the entire +amount of the sun’s heat which is continually lost by radiation, is +made up by the fall of meteors, a hypothesis which was propounded by +Mayer, and has been favourably adopted by several other physicists, is +open, according to Sir W. Thomson’s investigations, to objection; for, +assuming it to hold, the mass of the sun should increase so rapidly +that the consequences would have shown themselves in the accelerated +motion of the planets. The entire loss of heat from the sun cannot +at all events be produced in this way; at the most a portion, which, +however, may not be inconsiderable. + +If, now, there is no present manifestation of force sufficient to cover +the expenditure of the sun’s heat, the sun must originally have had a +store of heat which it gradually gives out. But whence this store? We +know that the cosmical forces alone could have produced it. And here +the hypothesis, previously discussed 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 capacity of the sun to +be the same as that of water, the temperature might be raised to +28,000,000 of degrees, if this quantity of heat could ever have been +present in the sun at one time. This cannot be assumed, for such +an increase of temperature would offer the greatest hindrance to +condensation. It is probable rather that a great part of this heat, +which was produced 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 condense +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 maintain for an additional +17,000,000 of years the same intensity of sunshine as that which is +now the source of all terrestrial life. + +The smaller bodies of our system might become less hot than the sun, +because the attraction of the fresh masses would be feebler. A body +like the earth might, if even we put its thermal capacity as high as +that of water, become heated to even 9,000 degrees, to more than our +flames can produce. The smaller bodies must cool more rapidly as long +as they are still liquid. The increase in temperature, with the depth, +is shown in bore-holes and in mines. The existence of hot wells and of +volcanic eruptions shows that in the interior of the earth there is a +very high temperature, which can scarcely be anything than a remnant +of the high temperature which prevailed at the time of its production. +At any rate, the attempts to discover for the internal heat of the +earth a more recent origin in chemical processes, have hitherto rested +on very arbitrary assumptions; and, compared with the general uniform +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, which characterises +larger masses.[26] 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. + +[Footnote 26: Mr. Zoellner concludes from photometric measurements, +which, however, need confirmation, that Jupiter still possesses a light +of its own.] + +[Illustration: FIG. 12.] + +You see, thus, by what various paths we are constantly 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 discoveries, by which the +conclusions are confirmed until we have confidence in them. In this +case another circumstance has contributed--that is, the observation +that this process of transformation, which the theory in question +presupposes, goes on still, though on a smaller scale, seeing that all +stages of that process can still be found to exist. + +[Illustration: FIG. 13.] + +For as we have already seen, the larger bodies which are already formed +go on increasing with the 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 firmament of fixed stars, according to Sir J. Herschel’s +newest catalogue, over 5,000 nebulous spots, of which those whose +light is sufficiently strong give for the most part a coloured spectrum +of fine bright lines, as they appear in the spectra of the ignited +gases. The nebulæ are partly rounded structures, which are called +_planetary nebulæ_ (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. + +[Illustration: FIG. 14.] + +[Illustration: FIG. 15.] + +[Illustration: FIG. 16.] + +In many nebulæ 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 nebulæ are only heaps of very fine stars, of other Milky Ways. +Now, however, spectrum analysis has shown a gas spectrum in many +nebulæ which contains stars, while actual heaps of stars show the +continuous spectrum of ignited solid bodies. Nebulæ have in general +three distinctly recognisable lines, one of which, in the blue, belongs +to hydrogen, a second in bluish-green to nitrogen,[27] while the third, +between the two, is of unknown origin. Fig. 17 shows 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 therefore +greatly enfeebled or even extinguished, while the undecomposable light +of bright gas lines remains undecomposed, and hence can still be seen. +In any case, the decomposition of the light of the nebulæ 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 +nebulæ exhibit highly irregular forms, which by no means correspond +to such a condition. As, however, their shape has either not at all +altered, or not appreciably, since they have been known and observed, +they must either have very little mass, or they must be of colossal +size and distance. The former does not appear very probable, because +small masses very soon give out their heat, and hence we are left to +the second alternative, that they are of huge dimensions and distances. +The same conclusion had been originally drawn by Sir W. Herschel, on +the assumption that the nebulæ were heaps of stars. + +[Footnote 27: Or perhaps also to oxygen. The line occurs in the +spectrum of atmospheric air, and according to H. C. Vogel’s +observations was wanting in the spectrum of pure oxygen.] + +[Illustration: FIG. 17.] + +With those nebulæ which, besides the lines of gases, also show the +continuous spectrum of ignited denser bodies, are connected spots which +are partly irresolvable and partly resolvable into heaps of stars, +which only show the light of the latter kind. + +The countless luminous stars of the heavenly firmament, 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 quantity of individual elements. + +But we find also in space a third stadium, that of extinct suns; +and for this also there are actual evidences. In the first place, +there are, in the course of history, pretty frequent examples of the +appearance of new stars. In 1572 Tycho Brahe observed such a one, +which, though gradually burning paler, was visible for two years, stood +still like a fixed star, and finally reverted to the darkness from +which it had 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 discovered themselves by +their attraction on adjacent bright stars, and the motions of the +latter thereby produced. 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 permit us to conclude that our sun also +is a body which slowly gives out its store of heat, and thus will some +time become extinct. + +The term of 17,000,000 years which I have given may perhaps become +considerably prolonged by the gradual abatement of radiation, by the +new accretion of falling meteors, and by still greater condensation +than that which I have assumed in that calculation. But we know of no +natural process which could spare our sun the fate which has manifestly +fallen upon other suns. This is a thought which we only reluctantly +admit; it seems to us an insult to the beneficent Creative Power which +we otherwise find at work in organisms and especially in living ones. +But we must reconcile ourselves to the thought that, however we may +consider ourselves to be the centre and final object of Creation, we +are but as dust on the earth; which again is but a speck of dust in +the immensity of space; and the previous duration of our race, even +if we follow it far beyond our written history, into the era of the +lake dwellings or of the mammoth, is but an instant compared with +the primeval times of our planet; when living beings existed upon +it, whose strange and unearthly remains still gaze at us from their +ancient tombs; and far more does the duration of our race sink into +insignificance compared with the enormous periods during which worlds +have been in 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? Considering 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 +equilibrium, will not consider 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 +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 dwelling-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 possibly 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 inheritance 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. But even +men of such free and large order of minds as Lessing and David Strauss +could not reconcile themselves 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 continues to exist--indestructible, imperishable, as an +invisible motion, now agitating the molecules of ponderable 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 +spectroscope. United afresh, these rays may ignite a new flame, and +thus, as it were, acquire a new bodily existence. + +Just as the flame remains the same in appearance, and continues to +exist with the same form and structure, 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 continuance of the individual is attached. For the material +of the body, like that of the flame, is subject to continuous 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? + + +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 further 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 +unscientific.” 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 Heidelberg and Cologne, but not published. I cannot +object if anyone considers this hypothesis to be in a high, or even in +the highest, degree improbable. But to me it seems a perfectly correct +scientific procedure, that when all our attempts fail in producing +organisms from inanimate matter, we may inquire whether life has +ever originated at all or not, and whether its germs have not been +transported from one world to another, and have developed themselves +wherever they found a favourable soil. + +‘Mr. Zoellner’s so-called physical objections are but of very small +weight. He recalls the history of meteoric stone, and adds (p. xxvi.): +“If, therefore, that meteoric stones covered with organisms had escaped +with a whole skin in the smash-up of its mother-body, and had not +shared the general rise of temperature, it must necessarily have first +passed through the atmosphere of the earth, before it could deliver +itself of its organisms for the purpose of peopling the earth.” + +‘Now, in the first place, we know from repeated observations 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 fragments, which at the first moment were scattered +with planetary 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. + +‘As I have already remarked I am not inclined to suggest that all these +possibilities are probabilities. They are questions the existence and +signification of which we must remember, in order that if the case +arise they may be solved by actual observations or by conclusions +therefrom.’ + + + + +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 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 developed. Much indeed has +been accomplished. + +Although all that we hoped for has not been fulfilled, and many things +have turned out differently from what we wished, yet we have gained +much for which we could not have dared to hope. Just as the history of +the world has made one of its few giant steps before our eyes, so also +has our science; hence an old student, like myself, scarcely recognises +the somewhat matronly aspect of Dame Medicine, when he accidentally +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 antagonism, 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 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 consulting +the literature of that period; they would meet with principles which +appear as if written in a lost tongue, so that it is by no means easy +for us to transfer ourselves into the mode of thought of a period +which is so far behind us. The course of development of medicine is an +instructive lesson on the true principles of scientific inquiry, and +the positive part of this lesson has, perhaps, in no previous time been +so impressively 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 generalisations, and has to discuss the +meaning of fundamental ideas; and which has, on that account, been not +unfitly termed Natural Philosophy by the English-speaking peoples. +Hence it does not fall too far out of the range of my official duties +and of my own studies, if I attempt to discourse here of the principles +of scientific method, in reference to the sciences of experience. + +As regards my acquaintance with the tone of thought of the older +medicine, independently of the general obligation, incumbent on every +educated physician, of understanding the literature of his science and +the direction as well as the conditions of its progress, there was in +my case a special incentive. In my first professorship at Königsberg, +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 +therapeutics, 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 home in +physical considerations found a fruitful virgin 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 importance 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 +commotion. Or he may consider ancient prejudices to be interesting and +pardonable, as remains of poetic romance, or of youthful enthusiasm. To +one who has to contend with the hostile forces of fact, indifference +and romance disappear; that which he knows and can do, is exposed to +severe tests; he can only use the hard and clear light of facts, and +must give up the notion of lulling himself in agreeable illusions. + +I rejoice, therefore, that I can once more address an assembly +consisting almost exclusively of medical men who have gone through the +same school. Medicine was once the intellectual home in which I grew +up, and even the emigrant best understands and is best understood by +his native land. + +If I am called upon to designate in one word the fundamental error +of that former time, I should be inclined 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 practically an exceedingly important +object of true philosophy. + +The proud word of Hippokrates, + + ἰητρὸς φιλόσοφος ἰσόθεος, + + ‘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 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 philosophical or +metaphysical considerations. If, therefore, we are to understand the +medical philosopher of Hippokrates 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 themselves 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 φιλόσοφος 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 completeness +of the system. It is not difficult to understand how in periods of +youthful development, such a one-sided over-estimate of thought could +be arrived at. The superiority of man over animals, of the scholar over +the barbarian, depends upon thinking; sensation, feeling, perception, +on the contrary, he shares with his lower fellow-creatures, and in +acuteness of the senses many of these are even superior to him. That +man strives to develop his thinking faculty to the utmost is a problem +on the solution of which the feeling of his own dignity, as well as of +his own practical power, depends; and it is a natural error to have +considered unimportant the dowry of mental capacities which Nature had +given to animals, and to have believed that thought could be liberated +from its natural basis, observation and perception, to begin its +Icarian flight of metaphysical speculation. + +It is, in fact, no easy problem to ascertain completely 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 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 contented 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 accustomed very often to mention together the corresponding +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 number 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, 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 everyday 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 conformity +with law is always repeated with regularity, these deeply impressed +remains of all previous conceptions are just the conceptions of what is +conformable to law in the things and processes. + +Thus man, when he begins to reflect, finds that he possesses a wide +range of acquirements of which he knows not whence they came, which he +has possessed as long as 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 +themselves, the world of facts seems to him, like his conceptions, 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 philosophy; 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, independently of their +theories. Then came the less gifted pupils, who copied their master, +exaggerated his theory, made it more one-sided and more logical, +without regard to any discordance with Nature. The more rigid the +system, the fewer and the more thorough were the methods to which the +healing art was restricted. The more the schools were driven into a +corner by the increase in actual knowledge, the more did they depend +upon the ancient authorities, and the more intolerant were they +against innovation. The great reformer of anatomy, Vesalius, was cited +before the Theological faculty of Salamanca; Servetus was burned at +Geneva along with his book, in which he described the circulation of +the lungs; and the Paris faculty prohibited the teaching of Harvey’s +doctrine of the circulation of the blood in its lecture rooms. + +At the same time the bases of the systems from which these schools +started were mostly views on natural science which it would have been +quite right to utilise within a narrow circle. What was not right was +the delusion that it was more scientific to refer all diseases to one +kind of explanation, than to several. What was called the solidar +pathology wanted to deduce everything from the altered mechanism of +the solid parts, especially from their altered tension; from the +_strictum_ and _laxum_, from tone and want of tone, and afterwards +from strained or relaxed nerves and from obstructions in the vessels. +Humoral pathology was only acquainted with alterations in mixture. +The four cardinal fluids, representatives of the classical four +elements, blood, phlegm, black and yellow gall; with others, the +acrimonies or dyscrasies, which had to be expelled by sweating and +purging; in the beginning of our modern epoch, the acids and alkalies +or the alchymistic spirits, and the occult qualities of the substances +assimilated--all these were the elements of this chemistry. Along with +these were found all kinds of physiological conceptions, some of which +contained remarkable foreshadowings, such as the ἔμφυτον θέρμον, 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 =πνεῦμα=, 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 +observation 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 disease, and their +cure, upon any one such simple explanation. Complete knowledge of the +causal connection of one class of phenomena gives finally a logical +coherent system. There is no prouder edifice of the most exact thought +than modern astronomy, deduced even to the minutest of its small +disturbances, from Newton’s law of gravitation. But Newton had been +preceded by Kepler, who had by induction collated all the facts; and +the astronomers have never believed that Newton’s force excluded the +simultaneous action of other forces. They have been continually on the +watch to see whether friction, resisting media, and swarms of meteors +have not also some influence. The older philosophers and physicians +believed they could deduce, before they had settled their general +principles by induction. They forgot that a deduction can have no more +certainty than the principle from which it is deduced; and that each +new induction must in the first place be a new test, by experience, of +its own bases. That a conclusion is deduced by the strictest logical +method from an uncertain premise 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 intolerance 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, +Aretæus, 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 _excitation_, and its capacity of responding to a +stimulus the _excitability_, which was lost at death. This entire +condition of things, which physically speaking asserts no 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 nourishment were considered to be the +normal stimuli. The _kind_ of activity seemed, on the contrary, to be +conditioned 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 treatment 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 the first half of the last +century, was professor of chemistry and pathology in Halle. Stahl had a +clear and acute mind, which is informing and stimulating, from the way +in which he states the proper question, even in those cases in which he +decides against our present views. He it is who established the first +comprehensive system of chemistry, that of phlogiston. If we translate +his phlogiston into latent heat, the theoretical bases of his system +passed essentially into the system of Lavoisier; Stahl did not then +know oxygen, which occasioned some false hypotheses; for instance, on +the negative gravity of phlogiston. Stahl’s ‘soul of life’ is, on the +whole, constructed on the pattern on which the pietistic communities +of that period represented to themselves the sinful human soul; it is +subject to errors and passions, to sloth, fear, impatience, sorrow, +indiscretion, despair. The physician must first appease it, or then +incite it, or punish it, and compel it to repent. And the way in which, +at the same time, he established the necessity of the physical and +vital actions was well thought out. The soul of life governs the body, +and only acts by means of the physico-chemical forces of the substances +assimilated. But it has the power to bind and to loose these forces, to +allow them full play or to restrain them. After death the restrained +forces become free, and evoke putrefaction and decomposition. For the +refutation of this hypothesis of binding and loosing, it was necessary +to discover the law of the conservation of force. + +The second half of the previous century was too much possessed by the +principles of rationalism to recognise openly Stahl’s ‘soul of life.’ +It was presented more scientifically as vital force, _Vis vitalis_, +while in the main it retained its functions, and under the name of +‘Nature’s healing power’ it played a prominent part in the treatment of +diseases. + +The doctrine of vital force entered into the pathological system of +changes in irritability. The attempt was made to separate the direct +actions of the virus which produce disease, in so far as they depended +on the play of blind natural forces, the _symptomata morbi_, from +those which brought on the reaction of vital force, the _symptomata +reactionis_. The latter were principally 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 according 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 treatment fell comparatively into the background. The +therapeutics of febrile diseases had thereby become very monotonous, +although the means indicated by theory were still abundantly used, +and especially blood-letting, which since that time has almost been +entirely abandoned. 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 essential 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 maintained that they were a coarse mechanical means of +investigation which a physician with a clear mental vision did not +need; and it indeed lowered and debased the patient, who was anyhow a +human being, by treating 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, however, 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 experiment. +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 +professor of therapeutics, who occupied himself much with the +reorganisation of the Universities, was urgent with 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 experienced, 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 foundation of the chemical laboratory, +was complete, as far as chemistry was concerned, but his example had +not been imitated elsewhere. Yet medicine possessed in anatomical +dissections a great means of education for independent observation, +which is wanting in the other faculties, and to which I am disposed +to attach great weight. Microscopic demonstrations were isolated +and infrequent in the lectures. Microscopic instruments were costly +and scarce. I came into possession of one by having spent my autumn +vacation in 1841 in the Charité, prostrated by typhoid fever; as pupil, +I was nursed without expense, and on my recovery 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 prolongations +of the ganglionic cells in the invertebrata, which I described in +my dissertation, and to investigate the vibrions in my research on +putrefaction and fermentation. + +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 exhaust all possibilities, in the +consideration, until a practicable path was found. We had, it is true, +an almost uncultivated field before us, in which almost every stroke of +the spade might produce remunerative results. + +It was one man more especially who aroused our enthusiasm for work in +the right direction--that is, Johannes Müller, 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 hypotheses, which had to be tested by +facts, and about which facts could alone decide. Even the views upon +those points which most easily crystallise into dogmas, on the mode of +activity of the vital force and the activity of the conscious soul, he +tried continually to define more precisely, to prove or to refute by +means of facts. + +And, although the art of anatomical investigation was most familiar to +him, and he therefore recurred most willingly to this, yet he worked +himself into the chemical and physical methods which were more foreign +to him. He furnished the proof that fibrine is dissolved in blood; he +experimented on the propagation of sound in such mechanisms as are +found in the drum of the ear; he treated the action of the eye as an +optician. His most important performance for the physiology of the +nervous system, as well as for the theory of cognition, was the actual +definite establishment of the doctrine of the specific energies of +the nerves. In reference to the separation of the nerves of motor and +sensible energy, he showed how to make the experimental proof of Bell’s +law of the roots of the spinal cord so as to be free from errors; and +in regard to the sensible energies he not only established the general +law, but carried out a great number of separate investigations, to +eliminate objections, and to refute false indications and evasions. +That which hitherto had been imagined from the data of everyday +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 Müller’s hands in a state of classical perfection--a +scientific achievement 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 example, 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, Brücke, Ludwig, Traube, J. Meyer, Lieberkühn, Hallmann; we +were succeeded by A. von Graefe, W. Busch, Max Schultze, A. Schneider. + +Microscopic and pathological anatomy, the study of organic types, +physiology, experimental pathology and therapeutics, ophthalmology, +developed themselves in Germany under the influence of this powerful +impulse far beyond the standard of rival adjacent countries. This was +helped by the labours of those of similar tendencies among Müller’s +contemporaries, among whom the three brothers Weber of Leipzig must +first of all be mentioned, who have built solid foundations in the +mechanism of the circulation, of the muscles, of the joints, and of the +ear. + +The attack was made wherever a way could be perceived of understanding +one of the vital processes; it was assumed that they could be +understood, and success justified this assumption. A delicate and +copious technical apparatus has been developed in the methods of +microscopy, of physiological chemistry, and of vivisection; the latter +greatly facilitated more particularly by the use of anæsthetic ether +and of the paralysing 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 possibilities of delicate and yet certain diagnosis +where there seemed to be absolute darkness. The continually increasing +number of proved parasitical organisms substitute 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 astounding conceit as to imagine that they +can effect, by a few clever strokes, that which man can otherwise only +hope to achieve by toilsome labour, hypotheses will be started which, +propounded as dogmas, at once promise to solve all riddles. And as long +as there are people who believe implicitly in that which they wish to +be true, so long will the hypotheses of the former find credence. Both +classes will certainly not die out, and to the latter the majority will +always belong. + +There are two characteristics more particularly which metaphysical +systems have always possessed. In the first place man is always +desirous of feeling himself to be a being of a higher order, far beyond +the standard of the rest of nature; this wish is satisfied by the +spiritualists. On the other hand, he would like to believe that by his +thought he was unrestrained lord of the world, and of course by his +thinking with those conceptions, to the development of which he has +attained; this is attempted to be satisfied by the 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. In order to acquire this foreknowledge of what is coming, +but of what has not been settled by observations, no other method is +possible than that of endeavouring to arrive at the laws of facts by +observations; 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 philosopher +reflects at each new phenomenon, whether the best established laws of +the best known forces may not experience a change; it can of course +only be a question of a change which does not contradict the whole +store of our previously collected experiences. It never thus attains +unconditional truth, but such a high degree of probability that it is +practically equal to certainty. The metaphysicians may amuse themselves +at this; we will take their mocking to heart when they are in a +position to do better, or even as well. The old words of Socrates, the +prime master of inductive definitions, in reference to them are just as +fresh as they were 2,000 years ago: ‘They imagined they knew what they +did not know, and he at any rate had the advantage of not pretending +to know what he did not know.’ And again, he was surprised at its not +being clear to them that it is not possible for men to discover such +things; since even those who most prided themselves on the speeches +made on the matter, did not agree among themselves, but behaved to each +other like madmen (τοῖς μαινομένοις ὁμοίως).[28] Socrates calls them +τοὺς μέγιστον φρονοῦντας. Schopenhauer[29] 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. + +[Footnote 28: Xenophon, _Memorabil._ I. i. 11.] + +[Footnote 29: Arthur Schopenhauer, _Von ihm. über ihn von Frauenstadt +und Lindner_. Berlin, 1863, p. 653.] + +In speaking against the empty manufacture of hypotheses, 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. 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 contemporaries. 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 developed--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 +perceptions. + +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 bringing 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 discovered; 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 hypotheses 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 hypotheses, has no value in +the progress of science. On the contrary, the few sound ideas which +they may contain are concealed by the rubbish of the rest; and one who +wants to publish something really new--facts--sees himself open to the +danger of countless claims of priority, unless he is prepared to waste +time and power in reading beforehand a quantity of absolutely useless +books, and to destroy his readers’ patience by a multitude of useless +quotations. + +Our generation has had to suffer under the tyranny of spiritualistic +metaphysics; the newer generation will probably have to guard against +that of the materialistic hypotheses. Kant’s rejection of the claims +of pure thought has gradually made some impression, but Kant allowed +one way of escape. It was as clear to him as to Socrates that all +metaphysical systems which up to that time had been propounded were +tissues of false conclusions. His _Kritik der reinen 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 metaphysics was striving after; and hence geometrical +axioms, which he looked upon as _à 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 _à +priori_ intuition has been the anchoring-ground of metaphysicians. 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 perception +of the senses is the expression of this theory in physiology. All +mathematicians united to fight against any attempt to resolve the +intuitions into their natural elements; whether the so-called pure or +the empirical, the axioms of geometry, the principles of mechanics, or +the perceptions of vision. For this reason, therefore, the mathematical +investigations of Lobatschewsky, Gauss, and Riemann on the alterations +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 experience--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 +investigators who endeavour to eliminate from among the perceptions of +the senses, whatever there may be of the actions of memory, and of the +repetition of similar impressions, 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 prolongation of ganglionic +cells, and of the connection of nerve-fibres in the brain. + +Indeed, however self-evident, and however important the principle may +appear to be, that natural science has to seek for the laws of facts, +this principle is nevertheless 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 explanation of phenomena. We cannot always +refer to the forces of atoms; we speak of a refractive force, of +electromotive 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 Müller gave +to the idea of reflex action, is gradually evaporated into this, that +when an impression 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 complexity of the nerve-fibres of the brain. But +the resemblance to the _qualitates occultæ_ of ancient medicine is very +suspicious. + +From the entire chain of my argument it follows that what I have +said against metaphysics is not intended against philosophy. But +metaphysicians have always tried to plume themselves on being +philosophers, and philosophical amateurs have mostly taken an interest +in the high-flying speculations of the metaphysicians, 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[30] I compared the +relationship 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 +connoisseurs 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. + +[Footnote 30: Preface to the German translation of Tyndall’s +_Scientific Fragments_, p. xxii.] + +In like manner, philosophy, if it gives up metaphysics, 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 insight 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 capabilities. The groping of the medical schools for the last +two thousand years is, among other things, an illustration 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 activity. 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 opponents, are usually not very polite +in their controversy; one’s own success may approximately be estimated +from the increasing want of politeness in the replies. + +My own researches have led me more than other disciples of the school +of natural science into controversial 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 warrantors to speak for +me--Socrates and Kant--both of whom were certain that all metaphysical +systems established up to their time were full of empty false +conclusions, and who guarded themselves against adding any new ones. In +order to show that the matter has not changed, either in the last 2,000 +years or in the last 100 years, let me conclude with a sentence of one +who was unfortunately too soon taken away from us, Frederick Albert +Lange, the author of the ‘History of Materialism.’ In his posthumous +‘Logical Studies,’ which he wrote in anticipation of his approaching +end, he gives the following picture, which struck me because it would +hold just as well in reference to solidar or humoral pathologists, or +any other of the old dogmatic schools of medicine. + +Lange says: The Hegelian ascribes to the Herbartian a less perfect +knowledge than to himself, and conversely; but neither hesitates to +consider the knowledge of the other to be higher compared with that of +the empiricist, and to recognise in it at any rate an approximation to +the only true knowledge. It is seen, also, that here no regard is paid +to the validity of the proof, and that a mere statement in the form of +a deduction from the entirety of a system is recognised as ‘apodictic +knowledge.’ + +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 nonsense 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 knowledge 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 epikrisis, 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. + + + + +ON ACADEMIC FREEDOM IN GERMAN UNIVERSITIES. + + +_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 necessitated many changes in the old +order of University teaching, and which will, perhaps, necessitate +other changes. It is indeed just in that branch (Physics) which I +represent, and which forms the theoretical basis of all other branches +of Natural Science, that the particular characteristics of their +methods are most definitely pronounced. I have already been several +times in the position of having to propose alterations 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 +thoughtless 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 Rector at Heidelberg, the +closest connection 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 prominently--that the real is far +more sharply separated from the unreal, by the incorruptible criticism +of facts, than is the case with the more complex problems of mental +science. + +And not merely the development of this new side of scientific activity, +which was almost unknown to antiquity, but also the influence of +many political, social, and even international relationships make +themselves felt, and require to be taken into account. The circle of +our students has had to be increased; a changed national life makes +other demands upon those who are leaving; the sciences become more and +more specialised and divided; exclusive of the libraries, larger and +more varied appliances for study are required. 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 prosperity of our Universities; we must +try to find what is the feature in their arrangements which we must +seek to retain as a precious jewel, and where, on the contrary, we +may give way when changes are required. I consider myself by no means +entitled to give a final opinion on this matter. The point of view of +any single individual 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 +degrees. The students of that time were mostly men of mature years, who +frequented the University more immediately for their own instruction +and without any direct practical object; but younger men soon began to +be sent, who, for the most part, were placed under the superintendence +of the older members. The separate Universities split again into closer +economic unions, under the name of ‘Nations,’ ‘Bursaries,’ ‘Colleges,’ +whose older members, the seniors, governed the common affairs of each +such union, and also met together for regulating the common affairs +of the University. In the courtyard of the University of Bologna are +still to be seen the coats-of-arms, and lists of members and seniors, +of many such Nations in ancient times. The older graduated members were +regarded as permanent life members of such Unions, and they retained +the right of voting, as is still the case in the College of Doctors +in the University of Vienna, and in the Colleges 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 compelled to quit +the University in a body. This was not restricted to those cases in +which the Church intermeddled, and where political or metaphysical +propositions were in question. Even the medical faculties--that of +Paris, the most celebrated of all at the head--allowed no divergence +from that which they regarded 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 Universities, Oxford and Cambridge. +Their great endowments, the political feeling of the English for the +retention 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 Roman and now of the Anglican +Church, whose instruction laymen might also share in so far as it could +serve the general education of the mind; they were subjected to such +a control and mode of life, as was formerly considered to be good for +young priests. They lived, as they still live, in colleges, under the +superintendence of a number of older graduate members (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 Repetitoria which are here +and there held in our Universities. 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 academical 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 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 cultivated 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 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, surrounded by lawns and groves of trees; they find +much of their pleasure in games which excite a passionate rivalry in +the development of bodily energy and skill, and which in this respect +are far more efficacious than our gymnastic and fencing exercises. It +must not be forgotten that the more young men are cut off from fresh +air and from the opportunity of vigorous exercise, the more induced +will they be to seek an apparent refreshment in the misuse of tobacco +and of intoxicating drinks. It must also be admitted that the English +Universities accustom their students to energetic and accurate work, +and keep them up to the habits of educated society. The _moral_ effect +of the more rigorous control is said to be rather illusory. + +The Scotch Universities and some smaller English foundations of more +recent origin--University College and King’s College in London, and +Owens College in Manchester--are constituted more on the German and +Dutch model. + +The development of French Universities has been quite different, and +indeed almost in the opposite direction. In accordance with the +tendency of the French to throw overboard everything of historic +development 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 distinct from those institutions which are to further the +progress of science, such as the _Collège de France_, the _Jardin des +Plantes_, and the _École des Études Supérieures_. 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 excite +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 instruction in France is +well adapted to give pupils, of even moderate capacity, sufficient +knowledge for the routine of their calling. They have no choice between +different teachers, and they swear _in verba magistri_; this gives a +happy self-satisfaction and freedom from doubts. If the teacher has +been well chosen, this is sufficient in ordinary cases, in which the +pupil does what he has seen his teacher do. It is only unusual cases +that test how much actual insight and judgment the pupil has acquired. +The French people are moreover gifted, vivacious, and ambitious, and +this corrects many defects 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 contribution towards the maintenance of the University. +When, therefore, the teacher has no real pleasure in teaching, or +is not ambitious of having a number of pupils, he very soon becomes +indifferent to the success of his teaching, and is inclined to take +things easily. + +Outside the lecture-rooms, the French students live without control, +and associate with young men of other callings, without any special +_esprit de corps_ or common feeling. + +The development of the German Universities differs characteristically +from these two extremes. Too poor in their own possessions not to be +compelled, with increasing demands for the means of instruction, +eagerly to accept the help of the State, and too weak to resist +encroachments upon their ancient rights in times in which modern +States attempt to consolidate themselves, 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 remained attached to it. It is +very remarkable how among wars and political changes in the States +fighting with the decaying Empire for the consolidation of their young +sovereignties, while almost all other privileged orders were destroyed, +the Universities of Germany saved a far greater nucleus of their +internal freedom and of the most valuable side of this freedom, than in +conscientious Conservative England, and than in France with its wild +chase after freedom. + +We have retained the old conception of students, as that of young men +responsible to themselves, striving after science of their own free +will, and to whom it is left to arrange their own plan of studies as +they think best. If attendance on particular lectures was enjoined +for certain callings--what are called ‘compulsory lectures’--these +regulations were not made by the University, 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 essential part of study. + +Outside the University there is no control over the proceedings of the +students, so long as they do not come in collision with the guardians +of public order. Beyond these cases the only control to which they are +subject is that of their colleagues, which prevents them from doing +anything which is repugnant to the feeling of honour of their own body. +The Universities of the Middle Ages formed definite close corporations, +with their own jurisdiction, which extended to the right over life +and death of their own members. As they lived for the most part on +foreign soil, it was necessary to have their own jurisdiction, partly +to protect the members from the caprices of foreign judges, partly to +keep up that degree of respect and order, within the society, which +was necessary to secure the continuation of the rights of hospitality +on a foreign soil; and partly, again, to settle disputes among the +members. In modern times the remains of this academic jurisdiction +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 guarantee 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 +considered fortunate that German students have retained a vivid sense +of corporate union, and of what is intimately connected therewith, +a requirement of honourable 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. + +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 expressions +of this feeling. Nothing of this kind is but even faintly suggested +in the literature of other European 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 Müller, the physiologist, made upon us, I must place a +very high value upon this latter point. Anyone who has once come in +contact with one or more men of the first rank must have had his whole +mental standard altered for the rest of his life. Such intercourse is, +moreover, the most interesting that life can offer. + +You, my younger friends, have received in this freedom of the German +students a costly and valuable inheritance of preceding generations. +Keep it--and hand it on to coming races, purified and ennobled if +possible. You have to maintain it, by each, in his place, taking care +that the body of German students is worthy of the confidence which has +hitherto accorded such a measure of freedom. But freedom necessarily +implies responsibility. It is as injurious a present for weak, as it is +valuable for strong characters. Do not wonder if parents and statesmen +sometimes urge that a more rigid system of supervision and control, +like that of the English, shall be introduced even among us. There +is no doubt that, by such a system, many a one would be saved who is +ruined by freedom. But the State and the Nation is best served by those +who can bear freedom, 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 instruction given, if possible, +only by teachers who have proved their own power of advancing science. +This also is a point in respect to which the English and French often +express their surprise. They lay more weight than the Germans on what +is called the ‘talent for teaching’--that is, the power of explaining +the subjects of instruction in a well-arranged and clear manner, and, +if possible, with eloquence, and so as to entertain and to fix the +attention. Lectures of eloquent orators at the Collège de France, +Jardin des Plantes, as well as in Oxford and Cambridge, are often the +centres of the elegant and the educated world. In Germany we are not +only indifferent to, but even distrustful of, oratorical ornament, +and often enough are more negligent than we should be of the outer +forms of the lecture. There can be no doubt that a good lecture can +be followed with far less exertion than a bad one; that the matter +of the first can be more certainly and completely apprehended; that +a well-arranged explanation, which develops the salient points and +the divisions of the subject, and which brings it, as it were, almost +intuitively before us, can impart more information in the same time +than one which has the opposite qualities. I am by no means prepared +to defend what is, frequently, our too great contempt for form in +speech and in writing. It cannot also be doubted that many original +men, who have done considerable scientific work, have often an uncouth, +heavy, and hesitating delivery. Yet I have not infrequently seen that +such teachers had crowded lecture-rooms, while empty-headed orators +excited astonishment in the first lecture, fatigue in the second, and +were deserted in the third. Anyone who desires to give his hearers a +perfect conviction of the truth of his principles must, first of all, +know from his own experience how conviction is acquired and how not. +He must have known how to acquire conviction where no predecessor had +been before him--that is, he must have worked at the confines of human +knowledge, and have conquered for it new regions. A teacher who retails +convictions which are foreign to him, is sufficient for those pupils +who depend upon authority as the source of their knowledge, but not for +such as require bases for their conviction which extend to the very +bottom. + +You will see that this is an honourable confidence which the nation +reposes in you. Definite courses and specified teachers are not +prescribed to you. You are regarded as men whose unfettered conviction +is to be gained; who know how to distinguish what is essential from +what is only apparent; who can no longer be appeased by an appeal to +any authority, and who no longer let themselves be so appeased. Care is +also always taken that you yourselves should penetrate to the sources +of knowledge in so far as these consist in books and monuments, or in +experiments, and in the observation of natural objects and processes. + +Even the smaller German Universities have their own libraries, +collections of casts, and the like. And in the establishment of +laboratories for chemistry, microscopy, physiology, and physics, +Germany has preceded all other European countries, who are now +beginning 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 consequences of materialistic metaphysics, the boldest +speculations upon the basis of Darwin’s theory of evolution, may be +taught in German Universities with as little restraint as the most +extreme deification of Papal Infallibility. As in the tribune of +European Parliaments 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 forbidden. 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 Collège de France the +lectures of a man of Renan’s scientific importance and earnestness are +forbidden. + +I have to speak of another aspect of our liberty of teaching. That is, +the extended sense in which German Universities have admitted teachers. +In the original meaning of the word, a doctor is a ‘teacher,’ or one +whose capacity as teacher is recognised. In the Universities of the +Middle Ages any doctor who found pupils could set up as teacher. In +course of time the practical signification of the title was changed. +Most of those who sought the title did not intend to act as teachers, +but only needed it as an official recognition of their scientific +training. Only in Germany are there any remains of this ancient right. +In accordance with the altered meaning of the title of doctor, and +the minuter specialisation of the subjects of instruction, a special +proof of more profound scientific proficiency, in the particular branch +in which they wish to habilitate, is required from those doctors who +desire to exercise the right of teaching. In most German Universities, +moreover, the legal status of these habilitated 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 +examination. 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 uncertain prospects for the future, +devote themselves to strenuous scientific work. And, judging us from +the point of view of basely practical interests, they are equally +surprised that the faculties so readily admit young men who at any +moment may change from assistants to competitors; and further, that +only in the most exceptional cases is anything ever heard of unworthy +means of competition in what is a matter of some delicacy. + +The appointment to vacant professorships, like the admission of +private docents, rests, though not unconditionally, and not in the +last resort, with the faculty, that is with the body of ordinary +professors. These form, in German Universities, that residuum of +former colleges of doctors to which the rights of the old corporations +have been transferred. They form as it were a select committee of the +graduates of a former epoch, but established with the co-operation +of the Government. The usual form for the nomination of new ordinary +professors is that the faculty proposes three candidates to Government +for its choice; where the Government, however, does not consider itself +restricted to the candidates proposed. Excepting in times of heated +party conflict it is very unusual for the proposals of the faculty to +be passed over. If there is not a very obvious reason for hesitation it +is always a serious personal responsibility for the executive officials +to elect, in opposition to the proposals of competent judges, a teacher +who has publicly to prove his capacity before large circles. + +The professors have, however, the strongest motives for securing +to the faculty the best teachers. The most essential condition for +being able to work with pleasure at the preparation of lectures is +the consciousness of having not too small a number of intelligent +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 successful 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 University 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 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 scientific 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 distinguished 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 personal interests. When the school +has scientifically out-lived itself, students will probably migrate by +degrees to other Universities. This may extend over a long period, and +the faculty in question will suffer during that time. + +We see best how strenuously the Universities under this system have +sought to attract the scientific ability of Germany when we consider +how many pioneers have remained outside the Universities. The answer +to such an inquiry is given in the not infrequent jest or sneer that +all wisdom in Germany is professorial wisdom. If we look at England, +we see men like Humphry Davy, Faraday, Mill, Grote, who have had +no connection with English Universities. If, on the other hand, we +deduct from the list of German men of science those who, like David +Strauss, have been driven away by Government 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 Royal Institution of London, a private +Society, which provides for its members 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 institutions which exert the most powerful attraction +on the taught. But it is clear that this attraction depends on the +teacher’s hope that he will not only find in the University a body of +pupils enthusiastic and accustomed 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 conviction, which is more +strongly impressed on the Germans than on their Aryan kindred of the +Celtic and Romanic branches, in whom practical political motives +have greater weight. They are able, and as it would seem with +perfect conscientiousness, to restrain the inquiring mind from the +investigation of those principles which appear to them to be beyond +the range of discussion, as forming the foundation of their political, +social, and religious organisation; they think themselves 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 considered as outside +the range of discussion, however remote and restricted it may be, and +however good may be the intention, the pupils must be kept in the +prescribed path, and teachers must be appointed who do not rebel +against authority. We can then, however, only speak of free conviction +in a very limited sense. + +You see how different was the plan of our forefathers. 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 conviction 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 Rome, also organised the German Universities. + +But any institution based upon freedom must also be able to calculate +on the judgment and reasonableness 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 necessitates 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 estimate which he places on his +task will depend on how far he is followed by the appreciation of a +sufficient 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 sufficiently logically trained +judgment, with a sufficient habit of mental exertion, with a tact +sufficiently developed 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 +themselves 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 frequented them, who were less developed in the above +respects. The general self-respect of the students 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 scrupulous 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 qualifications 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 independence of conviction. I say work; for independence +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 revolted for the right of +such conviction, and gave its witness in blood--is still in the van of +this fight. To Germany has fallen an exalted historical task, and in it +you are called upon to co-operate. + + + + +HERMANN VON HELMHOLTZ + +AN AUTOBIOGRAPHICAL SKETCH. + + +_An Address delivered on the occasion of his Jubilee, 1891._ + +In the course of the past year, and most recently on the occasion +of the celebration of my seventieth birthday, and the subsequent +festivities, I have been overloaded with honours, with marks of respect +and of goodwill in a way which could never have been expected. My +own sovereign, his Majesty the German Emperor, has raised me to the +highest rank in the Civil Service; the Kings of Sweden and of Italy, +my former sovereign, the Grand Duke of Baden, and the President of the +French Republic, have conferred Grand Crosses on me; many academies, +not only of science, but also of the fine arts, faculties, and learned +societies spread over the whole world, from Tomsk to Melbourne, have +sent me diplomas, and richly illuminated addresses, expressing in +elevated language their recognition of my scientific endeavours, and +their thanks for those endeavours, in terms which I cannot read without +a feeling of shame. My native town, Potsdam, has conferred its freedom +on me. To all this must be added countless individuals, scientific and +personal friends, pupils, and others personally unknown to me, who have +sent their congratulations in telegrams and in letters. + +But this is not all. You desire to make my name the banner, as it +were, of a magnificent institution which, founded by lovers of science +of all nations, is to encourage and promote scientific inquiry in +all countries. Science and art are, indeed, at the present time +the only remaining bond of peace between civilised nations. Their +ever-increasing development is a common aim of all; is effected by +the common work of all, and for the common good of all. A great and +a sacred work! The founders even wish to devote their gift to the +promotion of those branches of science which all my life I have +pursued, and thus bring me, with my shortcomings, before future +generations almost as an exemplar of scientific investigation. This +is the proudest honour which you could confer upon me, in so much as +you thereby show that I possess your unqualified favourable opinion. +But it would border on presumption were I to accept it without a quiet +expectation on my part that the judges of future centuries will not be +influenced by considerations of personal favour. + +My personal appearance even, you have had represented in marble by a +master of the first rank, so that I shall appear to the present and +to future generations in a more ideal form; and another master of the +etching needle has ensured that faithful portraits of me shall be +distributed among my contemporaries. + +I cannot fail to remember that all you have done is an expression of +the sincerest and warmest goodwill on your part, and that I am most +deeply indebted to you for it. + +I must, however, be excused if the first effect of these abundant +honours is rather surprising and confusing to me than intelligible. +My own consciousness does not justify me in putting a measure of the +value of what I have tried to do, which would leave such a balance +in my favour as you have drawn. I know how simply everything I have +done has been brought about; how scientific methods worked out by my +predecessors have naturally led to certain results, and how frequently +a fortunate circumstance or a lucky accident has helped me. But the +chief difference is this--that which I have seen slowly growing from +small beginnings through months and years of toilsome and tentative +work, all that suddenly starts before you like Pallas fully equipped +from the head of Jupiter. A feeling of surprise has entered into your +estimate, but not into mine. At times, and perhaps even frequently, +my own estimate may possibly have been unduly lowered by the fatigue +of the work, and by vexation about all kinds of futile steps which I +had taken. My colleagues, as well as the public at large, estimate a +scientific or artistic work according to the utility, the instruction, +or the pleasure which it has afforded. An author is usually disposed +to base his estimate on the labour it has cost him, and it is but +seldom that both kinds of judgment agree. It can, on the other hand, be +seen from incidental expressions of some of the most celebrated men, +especially of artists, that they lay but small weight on productions +which seem to us inimitable, compared with others which have been +difficult, and yet which appear to readers and observers as much less +successful. I need only mention Goethe, who once stated to Eckermann +that he did not estimate his poetical works so highly as what he had +done in the theory of colours. + +The same may have happened to me, though in a more modest degree, if I +may accept your assurances and those of the authors of the addresses +which have reached me. Permit me, therefore, to give you a short +account of the manner in which I have been led to the special direction +of my work. + +In my first seven years I was a delicate boy, for long confined to +my room, and often even to bed; but, nevertheless, I had a strong +inclination towards occupation and mental activity. My parents busied +themselves a good deal with me; picture books and games, especially +with wooden blocks, filled up the rest of the time. Reading came +pretty early, which, of course, greatly increased the range of my +occupations. But a defect of my mental organisation showed itself +almost as early, in that I had a bad memory for disconnected things. +The first indication of this I consider to be the difficulty I had +in distinguishing between left and right; afterwards, when at school +I began with languages, I had greater difficulties than others in +learning words, irregular grammatical forms, and peculiar terms of +expression. History as then taught to us I could scarcely master. To +learn prose by heart was martyrdom. This defect has, of course, only +increased, and is a vexation of my mature age. + +But when I possessed small mnemotechnical methods, or merely such as +are afforded by the metre and rhyme of poetry, learning by heart, +and the retention of what I had learnt, went on better. I easily +remembered poems by great authors, but by no means so easily the +somewhat artificial verses of authors of the second rank. I think that +is probably due to the natural flow of thought in good poems, and I am +inclined to think that in this connection is to be found an essential +basis of æsthetic beauty. In the higher classes of the Gymnasium I +could repeat some books of the Odyssey, a considerable number of the +odes of Horace, and large stores of German poetry. In other directions +I was just in the position of our older ancestors, who were not able +to write, and hence expressed their laws and their history in verse, so +as to learn them by heart. + +That which a man does easily he usually does willingly; hence I was +first of all a great admirer and lover of poetry. This inclination was +encouraged by my father, who, while he had a strict sense of duty, +was also of an enthusiastic disposition, impassioned for poetry, and +particularly for the classic period of German Literature. He taught +German in the upper classes of the Gymnasium, and read Homer with us. +Under his guidance we did, alternately, themes in German prose and +metrical exercises--poems as we called them. But even if most of us +remained indifferent poets, we learned better in this way, than in any +other I know of, how to express what we had to say in the most varied +manner. + +But the most perfect mnemotechnical help is a knowledge of the laws +of phenomena. This I first got to know in geometry. From the time +of my childish playing with wooden blocks, the relations of special +proportions to each other were well known to me from actual perception. +What sort of figures were produced when bodies of regular shape were +laid against each other I knew well without much consideration. When +I began the scientific study of geometry, all the facts which I had +to learn were perfectly well known and familiar to me, much to the +astonishment of my teachers. So far as I recollect, that came out +incidentally in the elementary school attached to the Potsdam Training +College, which I attended up to my eighth year. Strict scientific +methods, on the contrary, were new to me, and with their help I saw the +difficulties disappear which had hindered me in other regions. + +One thing was wanting in geometry; it dealt exclusively with abstract +forms of space, and I delighted in complete reality. As I became bigger +and stronger I went about with my father and my schoolfellows a great +deal in the neighbourhood of my native town, Potsdam, and I acquired +a great love of Nature. This is perhaps the reason why the first +fragments of physics which I learned in the Gymnasium engrossed me much +more closely than purely geometrical and algebraical studies. Here +there was a copious and multifarious region, with the mighty fulness +of Nature, to be brought under the dominion of a mentally apprehended +law. And, in fact, that which first fascinated me was the intellectual +mastery over Nature, which at first confronts us as so unfamiliar, +by the logical force of law. But this, of course, soon led to the +recognition that knowledge of natural processes was the magical key +which places ascendency over Nature in the hands of its possessor. In +this order of ideas I felt myself at home. + +I plunged then with great zeal and pleasure into the study of all +the books on physics I found in my father’s library. They were very +old-fashioned; phlogiston still held sway, and galvanism had not +grown beyond the voltaic pile. A young friend and myself tried, with +our small means, all sorts of experiments about which we had read. +The action of acids on our mothers’ stores of linen we investigated +thoroughly; we had otherwise but little success. Most successful was, +perhaps, the construction of optical instruments by means of spectacle +glasses, which were to be had in Potsdam, and a small botanical lens +belonging to my father. The limitation of our means had at that time +the value that I was compelled always to vary in all possible ways my +plans for experiments, until I got them in a form in which I could +carry them out. I must confess that many a time when the class was +reading Cicero or Virgil, both of which I found very tedious, I was +calculating under the desk the path of rays in a telescope, and I +discovered, even at that time, some optical theorems, not ordinarily +met with in text-books, but which I afterwards found useful in the +construction of the ophthalmoscope. + +Thus it happened that I entered upon that special line of study to +which I have subsequently adhered, and which, in the conditions I +have mentioned, grew into an absorbing impulse, amounting even to a +passion. This impulse to dominate the actual world by acquiring an +understanding of it, or what, I think, is only another expression for +the same thing, to discover the causal connection of phenomena, has +guided me through my whole life, and the strength of this impulse is +possibly the reason why I found no satisfaction in apparent solutions +of problems so long as I felt there were still obscure points in them. + +And now I was to go to the university. Physics was at that time looked +upon as an art by which a living could not be made. My parents were +compelled to be very economical, and my father explained to me that he +knew of no other way of helping me to the study of Physics, than by +taking up the study of medicine into the bargain. I was by no means +averse from the study of living Nature, and assented to this without +much difficulty. Moreover, the only influential person in our family +had been a medical man, the late Surgeon-General Mursinna; and this +relationship was a recommendation in my favour among other applicants +for admission to our Army Medical School, the Friedrich Wilhelms +Institut, which very materially helped the poorer students in passing +through their medical course. + +In this study I came at once under the influence of a profound +teacher--Johannes Müller; he who at the same time introduced E. +Du Bois-Reymond, E. Brücke, C. Ludwig, and Virchow to the study of +anatomy and physiology. As respects the critical questions about the +nature of life, Müller still struggled between the older--essentially +the metaphysical--view and the naturalistic one, which was then being +developed; but the conviction that nothing could replace the knowledge +of facts forced itself upon him with increasing certainty, and it may +be that his influence over his students was the greater because he +still so struggled. + +Young people are ready at once to attack the deepest problems, and thus +I attacked the perplexing question of the nature of the vital force. +Most physiologists had at that time adopted G. E. Stahl’s way out of +the difficulty, that while it is the physical and chemical forces of +the organs and substances of the living body which act on it, there +is an indwelling vital soul or vital force which could bind and loose +the activity of these forces; that after death the free action of +these forces produces decomposition, while during life their action is +continually being controlled by the soul of life. I had a misgiving +that there was something against nature in this explanation; but it +took me a good deal of trouble to state my misgiving in the form of a +definite question. I found ultimately, in the latter years of my career +as a student, that Stahl’s theory ascribed to every living body the +nature of a _perpetuum mobile_. I was tolerably well acquainted with +the controversies on this latter subject. In my school days I had +heard it discussed by my father and our mathematical teachers, and +while still a pupil of the Friedrich Wilhelms Institut I had helped in +the library, and in my spare moments had looked through the works of +Daniell, Bernouilli, D’Alembert, and other mathematicians of the last +century. I thus came upon the question, ‘What relations must exist +between the various kinds of natural forces for a perpetual motion to +be possible?’ and the further one, ‘Do those relations actually exist?’ +In my essay, ‘On the Conservation of Force,’ my aim was merely to give +a critical investigation and arrangement of the facts for the benefit +of physiologists. + +I should have been quite prepared if the experts had ultimately said, +‘We know all that. What is this young doctor thinking about, in +considering himself called upon to explain it all to us so fully?’ +But, to my astonishment, the physical authorities with whom I came in +contact took up the matter quite differently. They were inclined to +deny the correctness of the law, and in the eager contest in which +they were engaged against Hegel’s Natural Philosophy were disposed +to declare my essay to be a fantastical speculation. Jacobi, the +mathematician, who recognised the connection of my line of thought +with that of the mathematicians of the last century, was the only +one who took an interest in my attempt, and protected me from being +misconceived. On the other hand, I met with enthusiastic applause and +practical help from my younger friends, and especially from E. Du Bois +Reymond. These, then, soon brought over to my side the members of the +recently formed Physical Society of Berlin. About Joule’s researches on +the same subject I knew at that time but little, and nothing at all of +those of Robert Mayer. + +Connected with this were a few smaller experimental researches on +putrefaction and fermentation, in which I was able to furnish a proof, +in opposition to Liebig’s contention, that both were by no means purely +chemical decompositions, spontaneously occurring, or brought about by +the aid of the atmospheric oxygen; that alcoholic fermentation more +especially was bound up with the presence of yeast spores which are +only formed by reproduction. There was, further, my work on metabolism +in muscular action, which afterwards was connected with that on the +development of heat in muscular action; these being processes which +were to be expected from the law of the conservation of force. + +These researches were sufficient to direct upon me the attention of +Johannes Müller as well as of the Prussian Ministry of Instruction, +and to lead to my being called to Berlin as Brücke’s successor, and +immediately thereupon to the University of Königsberg. The Army medical +authorities, with thank-worthy liberality, very readily agreed to +relieve me from the obligation to further military service, and thus +made it possible for me to take up a scientific position. + +In Königsberg I had to lecture on general pathology and physiology. +A university professor undergoes a very valuable training in being +compelled to lecture every year, on the whole range of his science, in +such a manner that he convinces and satisfies the intelligent among his +hearers--the leading men of the next generation. This necessity yielded +me, first of all, two valuable results. + +For in preparing my course of lectures, I hit directly on the +possibility of the ophthalmoscope, and then on the plan of measuring +the rate of propagation of excitation in the nerves. + +The ophthalmoscope is, perhaps, the most popular of my scientific +performances, but I have already related to the oculists how luck +really played a comparatively more important part than my own merit. I +had to explain to my hearers Brücke’s theory of ocular illumination. In +this, Brücke was actually within a hair’s breadth of the invention of +the ophthalmoscope. He had merely neglected to put the question, To +what optical image do the rays belong, which come from the illuminated +eye? For the purpose he then had in view it was not necessary to +propound this question. If he had put it, he was quite the man to +answer it as quickly as I could, and the plan of the ophthalmoscope +would have been given. I turned the problem about in various ways, to +see how I could best explain it to my hearers, and I thereby hit upon +the question I have mentioned. I knew well, from my medical studies, +the difficulties which oculists had about the conditions then comprised +under the name of Amaurosis, and I at once set about constructing +the instrument by means of spectacle glasses and the glass used for +microscope purposes. The instrument was at first difficult to use, and +without an assured theoretical conviction that it must work, I might, +perhaps, not have persevered. But in about a week I had the great joy +of being the first who saw clearly before him a living human retina. + +The construction of the ophthalmoscope had a very decisive influence on +my position in the eyes of the world. From this time forward I met with +the most willing recognition and readiness to meet my wishes on the +part of the authorities and of my colleagues, so that for the future I +was able to pursue far more freely the secret impulses of my desire for +knowledge. I must, however, say that I ascribed my success in great +measure to the circumstance that, possessing some geometrical capacity, +and equipped with a knowledge of physics, I had, by good fortune, +been thrown among medical men, where I found in physiology a virgin +soil of great fertility; while, on the other hand, I was led by the +consideration of the vital processes to questions and points of view +which are usually foreign to pure mathematicians and physicists. Up to +that time I had only been able to compare my mathematical abilities +with those of my fellow-pupils and of my medical colleagues; that I was +for the most part superior to them in this respect did not, perhaps, +say very much. Moreover, mathematics was always regarded in the school +as a branch of secondary rank. In Latin composition, on the contrary, +which then decided the palm of victory, more than half my fellow-pupils +were ahead of me. + +In my own consciousness, my researches were simple logical applications +of the experimental and mathematical methods developed in science, +which by slight modifications could be easily adapted to the particular +object in view. My colleagues and friends, who, like myself, had +devoted themselves to the physical aspect of physiology, furnished +results no less surprising. + +But in the course of time matters could not remain in that stage. +Problems which might be solved by known methods I had gradually to +hand over, to the pupils in my laboratory, and for my own part turn to +more difficult researches, where success was uncertain, where general +methods left the investigator in the lurch, or where the method itself +had to be worked out. + +In those regions also which come nearer the boundaries of our knowledge +I have succeeded in many things experimental and mechanical--I do not +know if I may add philosophical. In respect of the former, like any one +who has attacked many experimental problems, I had become a person of +experience, who was acquainted with many plans and devices, and I had +changed my youthful habit of considering things geometrically into a +kind of mechanical mode of view. I felt, intuitively as it were, how +strains and stresses were distributed in any mechanical arrangement, +a faculty also met with in experienced mechanicians and machine +constructors. But I had the advantage over them of being able to make +complicated and specially important relations perspicuous, by means of +theoretical analysis. + +I have also been in a position to solve several mathematical physical +problems, and some, indeed, on which the great mathematicians, since +the time of Euler, had in vain occupied themselves; for example, +questions as to vortex motion and the discontinuity of motion in +liquids, the question as to the motion of sound at the open ends +of organ pipes, &c. &c. But the pride which I might have felt about +the final result in these cases was considerably lowered by my +consciousness that I had only succeeded in solving such problems +after many devious ways, by the gradually increasing generalisation +of favourable examples, and by a series of fortunate guesses. I had +to compare myself with an Alpine climber, who, not knowing the way, +ascends slowly and with toil, and is often compelled to retrace his +steps because his progress is stopped; sometimes by reasoning, and +sometimes by accident, he hits upon traces of a fresh path, which again +leads him a little further; and finally, when he has reached the goal, +he finds to his annoyance a royal road on which he might have ridden +up if he had been clever enough to find the right starting-point at +the outset. In my memoirs I have, of course, not given the reader an +account of my wanderings, but I have described the beaten path on which +he can now reach the summit without trouble. + +There are many people of narrow views, who greatly admire themselves, +if once in a way, they have had a happy idea, or believe they have +had one. An investigator, or an artist, who is continually having a +great number of happy ideas, is undoubtedly a privileged being, and +is recognised as a benefactor of humanity. But who can count or +measure such mental flashes? Who can follow the hidden tracts by which +conceptions are connected? + + That which man had never known, + Or had not thought out, + Through the labyrinth of mind + Wanders in the night. + +I must say that those regions, in which we have not to rely on lucky +accidents and ideas, have always been most agreeable to me, as fields +of work. + +But, as I have often been in the unpleasant position of having to +wait for lucky ideas, I have had some experience as to when and where +they came to me, which will perhaps be useful to others. They often +steal into the line of thought without their importance being at first +understood; then afterwards some accidental circumstance shows how +and under what conditions they have originated; they are present, +otherwise, without our knowing whence they came. In other cases they +occur suddenly, without exertion, like an inspiration. As far as my +experience goes, they never came at the desk or to a tired brain. +I have always so turned my problem about in all directions that I +could see in my mind its turns and complications, and run through +them freely without writing them down. But to reach that stage was +not usually possible without long preliminary work. Then, after the +fatigue from this had passed away, an hour of perfect bodily repose +and quiet comfort was necessary before the good ideas came. They often +came actually in the morning on waking, as expressed in Goethe’s words +which I have quoted, and as Gauss also has remarked.[31] But, as I have +stated in Heidelberg, they were usually apt to come when comfortably +ascending woody hills in sunny weather. The smallest quantity of +alcoholic drink seemed to frighten them away. + +[Footnote 31: Gauss, _Werke_, vol. v. p. 609. ‘The law of induction +discovered Jan. 23, 1835, at 7 A.M., before rising.’] + +Such moments of fruitful thought were indeed very delightful, but not +so the reverse, when the redeeming ideas did not come. For weeks or +months I was gnawing at such a question until in my mind I was + + Like to a beast upon a barren heath + Dragged in a circle by an evil spirit, + While all around are pleasant pastures green. + +And, lastly, it was often a sharp attack of headache which released me +from this strain, and set me free for other interests. + +I have entered upon still another region to which I was led by +investigation on perception and observation of the senses, namely, +the theory of cognition. Just as a physicist has to examine the +telescope and galvanometer with which he is working; has to get a clear +conception of what he can attain with them, and how they may deceive +him; so, too, it seemed to me necessary to investigate likewise the +capabilities of our power of thought. Here, also, we were concerned +only with a series of questions of fact about which definite answers +could and must be given. We have distinct impressions of the senses, +in consequence of which we know how to act. The success of the +action usually agrees with that which was to have been anticipated, +but sometimes also not, in what are called subjective impressions. +These are all objective facts, the laws regulating which it will be +possible to find. My principal result was that the impressions of the +senses are only signs for the constitution of the external world, the +interpretation of which must be learned by experience. The interest +for questions of the theory of cognition, had been implanted in me in +my youth, when I had often heard my father, who had retained a strong +impression from Fichter’s idealism, dispute with his colleagues who +believed in Kant or Hegel. Hitherto I have had but little reason to be +proud about those investigations. For each one in my favour, I have +had about ten opponents; and I have in particular aroused all the +metaphysicians, even the materialistic ones, and all people of hidden +metaphysical tendencies. But the addresses of the last few days have +revealed a host of friends whom as yet I did not know; so that in this +respect also I am indebted to this festivity for pleasure and for +fresh hope. Philosophy, it is true, has been for nearly three thousand +years the battle-ground for the most violent differences of opinion, +and it is not to be expected that these can be settled in the course of +a single life. + +I have wished to explain to you how the history of my scientific +endeavours and successes, so far as they go, appears when looked at +from my own point of view, and you will perhaps understand that I am +surprised at the universal profusion of praise which you have poured +out upon me. My successes have had primarily this value for my own +estimate of myself, that they furnished a standard of what I might +further attempt; but they have not, I hope, led me to self-admiration. +I have often enough seen how injurious an exaggerated sense of +self-importance may be for a scholar, and hence I have always taken +great care not to fall a prey to this enemy. I well knew that a +rigid self-criticism of my own work and my own capabilities was the +protection and palladium against this fate. But it is only needful +to keep the eyes open for what others can do, and what one cannot do +oneself, to find there is no great danger; and, as regards my own work, +I do not think I have ever corrected the last proof of a memoir without +finding in the course of twenty-four hours a few points which I could +have done better or more carefully. + +As regards the thanks which you consider you owe me, I should be unjust +if I said that the good of humanity appeared to me, from the outset, +as the conscious object of my labours. It was, in fact, the special +form of my desire for knowledge which impelled me and determined me, +to employ in scientific research all the time which was not required +by my official duties and by the care for my family. These two +restrictions did not, indeed, require any essential deviation from the +aims I was striving for. My office required me to make myself capable +of delivering lectures in the University; my family, that I should +establish and maintain my reputation as an investigator. The State, +which provided my maintenance, scientific appliances, and a great share +of my free time, had, in my opinion, acquired thereby the right that I +should communicate faithfully and completely to my fellow-citizens, and +in a suitable form, that which I had discovered by its help. + +The writing out of scientific investigations is usually a troublesome +affair; at any rate it has been so to me. Many parts of my memoirs I +have rewritten five or six times, and have changed the order about +until I was fairly satisfied. But the author has a great advantage in +such a careful wording of his work. It compels him to make the severest +criticism of each sentence and each conclusion, more thoroughly even +than the lectures at the University which I have mentioned. I have +never considered an investigation finished until it was formulated in +writing, completely and without any logical deficiencies. + +Those among my friends who were most conversant with the matter +represented to my mind, my conscience as it were. I asked myself +whether they would approve of it. They hovered before me as the +embodiment of the scientific spirit of an ideal humanity, and furnished +me with a standard. + +In the first half of my life, when I had still to work for my external +position, I will not say that, along with a desire for knowledge and a +feeling of duty as servant of the State, higher ethical motives were +not also at work; it was, however, in any case difficult to be certain +of the reality of their existence so long as selfish motives were +still existent. This is, perhaps, the case with all investigators. But +afterwards, when an assured position has been attained, when those who +have no inner impulse towards science may quite cease their labours, a +higher conception of their relation to humanity does influence those +who continue to work. They gradually learn from their own experience +how the thoughts which they have uttered, whether through literature +or through oral instruction, continue to act on their fellow-men, and +possess, as it were, an independent life; how these thoughts, further +worked out by their pupils, acquire a deeper significance and a more +definite form, and, reacting on their originators, furnish them with +fresh instruction. The ideas of an individual, which he himself has +conceived, are of course more closely connected with his mental field +of view than extraneous ones, and he feels more encouragement and +satisfaction when he sees the latter more abundantly developed than the +former. A kind of parental affection for such a mental child ultimately +springs up, which leads him to care and to struggle for the furtherance +of his mental offspring as he does for his real children. + +But, at the same time, the whole intellectual world of civilised +humanity presents itself to him as a continuous and spontaneously +developing whole, the duration of which seems infinite as compared +with that of a single individual. With his small contributions to the +building up of science, he sees that he is in the service of something +everlastingly sacred, with which he is connected by close bands of +affection. His work thereby appears to him more sanctified. Anyone can, +perhaps, apprehend this theoretically, but actual personal experience +is doubtless necessary to develop this idea into a strong feeling. + +The world, which is not apt to believe in ideal motives, calls this +feeling love of fame. But there is a decisive criterion by which both +kinds of sentiment can be discriminated. Ask the question if it is +the same thing to you whether the results of investigation which you +have obtained are recognised as belonging to you or not when there +are no considerations of external advantage bound up with the answer +to this question. The reply to it is easiest in the case of chiefs +of laboratories. The teacher must usually furnish the fundamental +idea of the research as well as a number of proposals for overcoming +experimental difficulties, in which more or less ingenuity comes into +play. All this passes as the work of the student, and ultimately +appears in his name when the research is finished. Who can afterwards +decide what one or the other has done? And how many teachers are there +not who in this respect are devoid of any jealousy? + +Thus, gentlemen, I have been in the happy position that, in freely +following my own inclination, I have been led to researches for which +you praise me, as having been useful and instructive. I am extremely +fortunate that I am praised and honoured by my contemporaries, in so +high a degree, for a course of work which is to me the most interesting +I could pursue. But my contemporaries have afforded me great and +essential help. Apart from the care for my own existence and that of +my family, of which they have relieved me, and apart from the external +means with which they have provided me, I have found in them a standard +of the intellectual capacity of man; and by their sympathy for my work +they have evoked in me a vivid conception of the universal mental life +of humanity which has enabled me to see the value of my own researches +in a higher light. In these circumstances, I can only regard as a free +gift the thanks which you desire to accord to me, given unconditionally +and without counting on any return. + + PRINTED BY + SPOTTISWOODE AND CO. LTD., NEW STREET SQUARE + LONDON + + + WORKS BY JOHN TYNDALL, D.C.L. LL.D. F.R.S. + + FRAGMENTS of SCIENCE: a Series of Detached Essays, + Addresses, and Reviews. 2 vols. crown 8vo. + 16_s._ + + NEW FRAGMENTS. Crown 8vo. 10_s._ 6_d._ + + LECTURES on SOUND. With Frontispiece of Fog-Syren, and + 203 other Woodcuts and Diagrams in the Text. Crown + 8vo. 10_s._ 6_d._ + + HEAT, a MODE of MOTION. With 125 Woodcuts and Diagrams. + Crown 8vo. 12_s._ + + LECTURES on LIGHT, DELIVERED in the UNITED STATES in + 1872 and 1873. With Portrait, Lithographic Plate, + and 59 Diagrams. Crown 8vo. 5_s._ + + ESSAYS on the FLOATING MATTER of the AIR in RELATION to + PUTREFACTION and INFECTION. With 24 Woodcuts. 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Fcp. 8vo. + 3_s._ 6_d._ + + LONGMANS, GREEN, & CO. 39 Paternoster Row, London + New York, Bombay, and Calcutta. +*** END OF THE PROJECT GUTENBERG EBOOK 77725 *** diff --git a/77725-h/77725-h.htm b/77725-h/77725-h.htm new file mode 100644 index 0000000..4ef86cc --- /dev/null +++ b/77725-h/77725-h.htm @@ -0,0 +1,7418 @@ +<!DOCTYPE html> +<html lang="en"> +<head> + <meta charset="UTF-8"> + <meta name="viewport" content="width=device-width, initial-scale=1"> + <title> + Popular lectures on scientific subjects | Project Gutenberg + </title> + <link rel="icon" href="images/cover.jpg" type="image/x-cover"> + <style> + +body { margin-left: 10%; margin-right: 10%; } + +h1,h2,h3,h4,h5,h6 { text-align: center; clear: both; } +.h_subtitle{font-weight: normal; font-size: smaller;} + + +p { margin-top: .51em; text-align: justify; text-indent: 1.5em; margin-bottom: .49em; } +p.no-indent { margin-top: .51em; text-align: justify; text-indent: 0em; margin-bottom: .49em;} +p.author { margin-top: 1em; margin-right: 5%; text-align: right;} +p.neg-indent { text-indent: -1.5em; margin-left: 5%; margin-right: 5%; padding-left: 1.5em;} + +p.f90 { font-size: 90%; text-align: center; text-indent: 0em; } +p.f110 { font-size: 110%; text-align: center; text-indent: 0em; } +p.f120 { font-size: 120%; text-align: center; text-indent: 0em; } +p.f150 { font-size: 150%; text-align: center; text-indent: 0em; } + +.fs_80 { font-size: 80%; } +.fs_110 { font-size: 110%; } +.fs_120 { font-size: 120%; } +.fs_150 { font-size: 150%; } +.fs_200 { font-size: 200%; } + +.spa1 { margin-top: 1em; } +.spa2 { margin-top: 2em; } +.spb1 { margin-bottom: 1em; } +.spb2 { margin-bottom: 2em; } + +hr { width: 80%; margin-top: 2em; margin-bottom: 2em; margin-left: 10%; margin-right: 10%; clear: both; } +hr.tb {width: 45%; margin-left: 27.5%; margin-right: 27.5%; margin-top: 1.5em; margin-bottom: 1.5em;} +hr.chap {width: 65%; margin-left: 17.5%; margin-right: 17.5%; margin-top: 2em; margin-bottom: 2em;} + @media print { hr.chap {display: none; visibility: hidden;} } +hr.r10 {width: 10%; margin-top: 0.5em; margin-bottom: 0.5em; margin-left: 45%; margin-right: 45%;} + +div.chapter {page-break-before: always;} +h2.nobreak {page-break-before: avoid;} + +ul.index { list-style-type: none; white-space: nowrap; } +li.isub2 {text-indent: 2em;} +li.isub3 {text-indent: 3em;} +li.isub5 {text-indent: 5em;} +li.isub6 {text-indent: 6em;} +li.isub7 {text-indent: 7em;} +li.isub9 {text-indent: 9em;} + +table { margin-left: auto; margin-right: auto; white-space: nowrap; + border-spacing: 0px; border-collapse: collapse; } + +th, td { padding-top: 0px; padding-bottom: 0px; + padding-left: 0px; padding-right: 0px; } + +.tdl {text-align: left;} +.tdr {text-align: right;} +.tdc {text-align: center;} +.tdc_wsp {text-align: center; vertical-align: middle; padding-left: 0.5em; padding-right: 0.5em;} +.tdl_ws1 {text-align: left; vertical-align: middle; padding-left: 1em;} +.tdl_ws2 {text-align: left; vertical-align: middle; padding-left: 2em;} +.tdr_wsp {text-align: right; vertical-align: middle; padding-right: 0.5em;} + +.pagenum { + position: absolute; + left: 92%; + font-size: small; + text-align: right; + font-style: normal; + font-weight: normal; + font-variant: normal; + text-indent: 0; +} + +.blockquot { margin-top: 0; margin-bottom: 0; + margin-left: 12%; margin-right: 12%; } + +.bb {border-bottom: solid thin;} +.bbox {border: solid medium;} + +.no-wrap {white-space: nowrap; } +.center {text-align: center; text-indent: 0em;} +.smcap {font-variant: small-caps;} +.allsmcap {font-variant: small-caps; text-transform: lowercase;} +.over {text-decoration: overline;} + +img { max-width: 100%; height: auto; } + +.figcenter { margin: auto; + text-align: center; + page-break-inside: avoid; + max-width: 100%; +} + +div.figcontainer { clear: both; margin: 0em auto; text-align: center; max-width: 100%;} +div.figsub { display: inline-block; margin: 1em 1em; vertical-align: top; + max-width: 100%; text-align: center; } + +.footnotes {border: 1px dashed;} +.footnote {margin-left: 10%; margin-right: 10%; font-size: 0.9em;} +.footnote .label {position: absolute; right: 84%; text-align: right;} +.fnanchor { vertical-align: super; + font-size: .8em; + text-decoration: + none; +} + +.transnote { background-color: #E6E6FA; + color: black; + font-size:small; + padding:0.5em; + margin-bottom:5em; + font-family:sans-serif, serif; } + +.ws2 {display: inline; margin-left: 0em; padding-left: 2em;} +.ws3 {display: inline; margin-left: 0em; padding-left: 3em;} +.ws4 {display: inline; margin-left: 0em; padding-left: 4em;} + + </style> +</head> +<body> +<div style='text-align:center'>*** START OF THE PROJECT GUTENBERG EBOOK 77725 ***</div> + +<hr class="chap x-ebookmaker-drop"> + +<h1>POPULAR LECTURES<br><span class="fs_80">ON</span><br>SCIENTIFIC SUBJECTS</h1> + +<p class="f150"><span class="fs_80">BY</span><br>HERMANN VON HELMHOLTZ</p> + +<p class="f120">TRANSLATED BY<br><span class="smcap">E. ATKINSON, Ph.D.</span>, F.C.S.</p> + +<p class="f90">FORMERLY PROFESSOR OF EXPERIMENTAL SCIENCE,<br> STAFF COLLEGE</p> + +<p class="f120">SECOND SERIES</p> + +<p class="center"><i>WITH AN AUTOBIOGRAPHY OF THE AUTHOR</i></p> + +<div class="figcenter"> + <img src="images/logo.jpg" alt="" width="200" height="265" > +</div> + +<p class="f120">LONGMANS, GREEN, AND CO.</p> +<p class="center">39 PATERNOSTER ROW, LONDON<br>NEW YORK, BOMBAY, AND CALCUTTA<br> +1908</p> + +<p class="center"><i>All rights reserved</i></p> + +<p class="f120 spa2"><i>BIBLIOGRAPHICAL NOTE.</i></p> + +<p class="center"><i>Issued in Silver Library, March 1893; Reprinted<br> +July 1895, May 1898, June 1900, December 1903,<br>August 1908.</i></p> + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_v">[Pg v]</span></p> + <p class="f150"><b>PREFACE.</b></p> +</div> + +<p>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.</p> + +<p>I have to express my acknowledgments to Professor G. Croome Robertson, +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.</p> + +<p>The article on ‘Academic Freedom in German Universities’ contains some +statements respecting the Universities of Oxford and Cambridge to which +exception 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 +<span class="pagenum" id="Page_vi">[Pg vi]</span> +same extent. The reform in the University system, which may be said to +date from the year 1854, has brought about so many alterations both in +the form and in the spirit of the regulations, that older members of +the University have been known to speak of the place as so changed that +they could scarcely recognise it. Hence, in respect of this article, +I have availed myself of the liberty granted by Professor Helmholtz, +and have altogether omitted some passages, and have slightly modified +others, which would convey an erroneous 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.</p> + +<p>In other articles, where the matter is of prime importance, I have been +anxious faithfully to reproduce the original; nor have I in any such +cases allowed a regard for form to interfere with the plain duty of +exactly rendering the author’s meaning.</p> + +<p class="author">E. ATKINSON.</p> + +<p><span class="smcap">Portesbery Hill, Camberley</span>:<br> + <span class="ws3"><i>Dec. 1880</i>.</span></p> + +<p><span class="pagenum" id="Page_vii">[Pg vii]</span></p> + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p class="f150"><b>CONTENTS.</b></p> +</div> + +<table class="spb1 smcap"> + <tbody><tr> + <td class="tdl fs_80">LECTURE</td> + <td class="tdc"> </td> + <td class="tdr fs_80">PAGE</td> + </tr><tr> + <td class="tdr_wsp">I.</td> + <td class="tdl">Gustav Magnus. In Memoriam</td> + <td class="tdr_wsp"><a href="#Page_1">1</a></td> + </tr><tr> + <td class="tdr_wsp">II.</td> + <td class="tdl">On the Origin and Significance of Geometrical Axioms  </td> + <td class="tdr_wsp"><a href="#Page_27">27</a></td> + </tr><tr> + <td class="tdr_wsp">III.</td> + <td class="tdl">On the Relation of Optics to Painting</td> + <td class="tdr_wsp"><a href="#Page_73">73</a></td> + </tr><tr> + <td class="tdl"> </td> + <td class="tdl_ws1">   I. Form</td> + <td class="tdr_wsp"><a href="#Page_78">78</a></td> + </tr><tr> + <td class="tdl"> </td> + <td class="tdl_ws1">  II. Shade</td> + <td class="tdr_wsp"><a href="#Page_94">94</a></td> + </tr><tr> + <td class="tdl"> </td> + <td class="tdl_ws1">III. Colour</td> + <td class="tdr_wsp"><a href="#Page_110">110</a></td> + </tr><tr> + <td class="tdl"> </td> + <td class="tdl_ws1">IV. Harmony of Colour</td> + <td class="tdr_wsp"><a href="#Page_124">124</a></td> + </tr><tr> + <td class="tdr_wsp">IV.</td> + <td class="tdl">On the Origin of the Planetary System</td> + <td class="tdr_wsp"><a href="#Page_139">139</a></td> + </tr><tr> + <td class="tdr_wsp">V.</td> + <td class="tdl">On Thought in Medicine</td> + <td class="tdr_wsp"><a href="#Page_199">199</a></td> + </tr><tr> + <td class="tdr_wsp">VI.</td> + <td class="tdl">On Academic Freedom in German Universities</td> + <td class="tdr_wsp"><a href="#Page_237">237</a></td> + </tr><tr> + <td class="tdr_wsp">VII.</td> + <td class="tdl">Hermann von Helmholtz.</td> + <td class="tdr_wsp"> </td> + </tr><tr> + <td class="tdr_wsp"> </td> + <td class="tdl_ws2">An Autobiographical Sketch</td> + <td class="tdr_wsp"><a href="#Page_266">266</a></td> + </tr> + </tbody> +</table> + +<p><span class="pagenum"><a id="Page_viii">[Pg viii]</a></span></p> + +<hr class="chap x-ebookmaker-drop"> +<p><span class="pagenum"><a id="Page_1">[Pg 1]</a></span></p> +<div class="chapter"> + <h2 class="nobreak">GUSTAV MAGNUS.<br><span class="h_subtitle"><i>In Memoriam.</i></span></h2> +</div> + +<p class="center"><i>Address delivered in the Leibnitz meeting of<br>the Academy of Sciences +on July 6, 1871.</i></p> + +<p>The honourable duty has fallen on me of expressing in the name of this +Academy what it has lost in Gustav Magnus, who belonged to it for +thirty years. As a grateful pupil, as a friend, and finally as his +successor, it was a pleasure to me as well as a duty to fulfil such a +task. But I find the best part of my work already done by our colleague +Hofmann at the request of the German Chemical Society, of which he is +the President. He has solved the difficulty of giving a picture of the +life and work of Magnus in the most complete 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 +<span class="pagenum" id="Page_2">[Pg 2]</span> +hand, he is much better qualified than I to pronounce a competent +judgment on the principal side of Magnus’s activity, namely, the +chemical.</p> + +<p>Hence what remains for me to do is greatly restricted. 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.</p> + +<p>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. Heinrich Gustav Magnus was born in Berlin on May 2, 1802, +the fourth of six brothers, who by their talents have distinguished +themselves in various directions. The father, Johann Matthias, was +chief of a wealthy commercial house, whose first concern was to secure +to his children a free development of their individual capacity +and inclinations. Our departed friend showed very early a greater +inclination for the study of mathematics and natural philosophy than +for that of languages. His father arranged his instruction accordingly, +by removing him from the Werder Gymnasium and sending him to the Cauer +Private Institute, in which more attention was paid to scientific subjects. +<span class="pagenum" id="Page_3">[Pg 3]</span></p> + +<p>From 1822 to 1827 Magnus devoted himself entirely 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 Dulong, Thénard 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 professor 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.</p> + +<p>His life was passed thus in quiet but unremitting activity; travels, +sometimes for scientific or technical studies, sometimes also in the +<span class="pagenum" id="Page_4">[Pg 4]</span> +service of the State, and occasionally for recreation, interrupted +his work here from time to time. His experience and knowledge of +business were often in demand by the State on various commissions; +among these may be especially mentioned the part he took in the +chemical deliberations of the Agricultural Board (<i>Landes-Economie +Collegium</i>), to which he devoted much of his time; above all to the +great practical questions of agricultural chemistry.</p> + +<p>After sixty-seven years of almost undisturbed health he was overtaken +by a painful illness towards the end of the year 1869.⁠<a id="FNanchor_1_1" href="#Footnote_1_1" class="fnanchor">[1]</a> +He still continued his lectures on physics until February 25, 1870, but during +March he was scarcely able to leave his bed, and he died on April 4.</p> + +<p>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 +<span class="pagenum" id="Page_5">[Pg 5]</span> +with the happy few among 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.</p> + +<p>Accustomed from an early age to the regulated and prudent activity of +the commercial house in which he grew up, he retained that skill in +business which he had so frequently to exercise in the administration +of the affairs of this Academy, of the philosophical faculty, and of +the various Government commissions. He retained from thence the love of +order, the tendency towards the actual, and towards what is practically +attainable, even although the chief aim of his activity was an ideal +one. He understood that the pleasant enjoyment of an existence free +from care, and intercourse with the most amiable circle of relatives +and friends, do not bring a lasting satisfaction; but work only, and +unselfish work for an ideal aim. Thus he laboured, not for the increase +of riches, but for science; not as a dilettante and capriciously, but +according to a fixed aim and indefatigably; not in vanity, catching +at striking discoveries, which might at once have made his name +<span class="pagenum" id="Page_6">[Pg 6]</span> +celebrated. He was, on the contrary, a master of faithful, 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 simultaneously, 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.</p> + +<p>The unselfishness with which Magnus held to the ideal aim of his +efforts is shown in quite a characteristic 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 appliances 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. +<span class="pagenum" id="Page_7">[Pg 7]</span></p> + +<p>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.</p> + +<p>By such a course he not only surrendered the external 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.</p> + +<p>In such circumstances we see an essential part of the inner tendency of +<span class="pagenum" id="Page_8">[Pg 8]</span> +the man, which must not be 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.</p> + +<p>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 apparatus 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 recollect very well with what wonder and +admiration we students saw him experiment, not merely because all the +experiments were successful and brilliant, but because they scarcely +<span class="pagenum" id="Page_9">[Pg 9]</span> +seemed to occupy or to disturb his thoughts. The easy and clear flow of +his discourse went on without interruption; each experiment came in its +right place, was performed quickly, without haste or hesitation, and +was then put aside.</p> + +<p>I have already mentioned that the valuable collection of apparatus came +into the possession of the University during his lifetime. He specially +wished that what he had collected and constructed as appliances 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 +laboratory, as well as his very rich and valuable library, and he thus +laid the foundation for the further development of a Public Physical +Institute.</p> + +<p>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.</p> + +<p>Personal recollections will furnish a livelier image to all those of +you who have worked with him for the last thirty years.</p> + +<p>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 +<span class="pagenum" id="Page_10">[Pg 10]</span> +to his fellow-creatures. To this must be added, that he lived in an +age when natural science passed through a process of development, with +a rapidity such as never occurred before in the history of science. +But the men who belonged to such a time, and co-operated in this +development are apt to appear in wrong perspective to their successors, +since the best part of their work seems to the latter self-evident, and +scarcely worthy of mention.</p> + +<p>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 possession, 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 revolutionising +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.</p> + +<p>It is strange to see how late and with what hesitation Germans applied +<span class="pagenum" id="Page_11">[Pg 11]</span> +themselves to the study of natural science in this century, whilst they +had taken so prominent a part in its earlier development. I need only +name Copernicus, Kepler, Leibnitz, and Stahl.</p> + +<p>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 principle to their +ultimate sources, and to trouble ourselves but little about what has no +connection with fundamental principles, and especially about practical +consequences 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, +politically 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 intellectual life, for all this there was no time during the +<span class="pagenum" id="Page_12">[Pg 12]</span> +storm of war; each man had to hold to his own party, every incipient +change of opinion was looked upon as treachery, and excited bitter +wrath. Owing to the Reformation, intellectual life had lost its old +stability and cohesion; everything appeared in a new light, and new +questions arose. The German mind could not be quieted with outward +uniformity; when it was not convinced and satisfied, it did not allow +its doubt to remain silent. Thus it was theology, and next to it +classical philology and philosophy, which, partly as scientific aids +of theology, partly for what they could do for the solution of the +new moral, æsthetical, and metaphysical problems, laid claim almost +exclusively to the interest of scientific culture. Hence it is clear +why the Protestant nations, as well as that part of the Catholics +which, wavering in its old faith, only remained outwardly in connection +with its church, threw itself with such zeal on philosophy. Ethical +and metaphysical problems 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 enumerate the actual results +which the last century gained by this work. It excited soaring hopes, +and it cannot be denied that metaphysics has a dangerous attraction +for the German mind; it could not again abandon it until all its +hiding-places had been searched through and it had satisfied itself +that for the present nothing more is to be found there. +<span class="pagenum" id="Page_13">[Pg 13]</span></p> + +<p>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 transformed itself into one of the most +expressive instruments of the human mind; out of what was still the +hard, poor, and wearisome condition of civil and political life, the +results of the religious war, in which the figure of the Prussian +hero-king only now cast the first hope of a better future, to be again +followed by the misery of the Napoleonic war,—out of this joyless +existence, all sensitive minds gladly fled into the flowery land opened +out by German poetry, rivalling as it did the best poetry of all times +and of all peoples; or in the sublime aspects of philosophy they +endeavoured to sink reality in oblivion.</p> + +<p>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 +<span class="pagenum" id="Page_14">[Pg 14]</span> +those natural philosophers who, like Oken, rejoiced in poetical +philosophical conceptions, who found willing auditors.</p> + +<p>Far be it from me as a one-sided advocate of scientific interests to +blame this period of enthusiastic excitement; 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 resource 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.</p> + +<p>Against the errors of a mental tendency, which corresponded at first to +the natural soaring of a fresh youthful ambition, but which afterwards, +in the age of the Epigones of the Romantic school and of the philosophy +of Identity, fell into sentimental straining after sublimity and +inspiration, a reaction took place, and was carried out not merely in +the regions of science, but also in history, in art, and in philology. +In the last departments, too, where we deal directly with products of +<span class="pagenum" id="Page_15">[Pg 15]</span> +activity of the human mind, and where, therefore, a construction <i>à +priori</i> from the psychological laws seems much more possible than in +nature, it has come to be understood that we must first know the facts, +before we can establish their laws.</p> + +<p>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 endeavoured 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 continually preached +to his pupils that no reasoning, however plausible it might seem, +avails against actual fact, and that observation and experiment must +decide; and he was always anxious that every practicable experiment +should be made which could give practical confirmation or refutation +of an assumed law. He did not limit in any way the applicability of +scientific methods in the investigation of inanimate nature, but in his +<span class="pagenum" id="Page_16">[Pg 16]</span> +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 respiration; +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.</p> + +<p>He cannot be reproached with having had too <i>little</i> 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 mathematical 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 standpoint +which science has now attained, it must be confessed that his +distrust of the mathematical physics of that date was not unfounded. +At that time no separation 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 elements which formed the basis +of calculation was put forth as axioms of metaphysical necessity, +and postulated a similar kind of necessity for the results. I need +only recall to you the great part which hypotheses as to the atomic +<span class="pagenum" id="Page_17">[Pg 17]</span> +structure of bodies played in mathematical physics during the first +half of this century, whilst as good as nothing was known of atoms; +and, for instance, hardly anything was known of the extraordinary +influence which heat has on molecular 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.</p> + +<p>In reference to atoms in molecular physics, Sir W. Thomson says, with +much weight, that their assumption 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 mathematicians who in England followed Faraday’s +lead, Stokes, W. Thomson, and Clerk-Maxwell. It is now understood that +mathematical physics is a purely experimental science; that it has no +other principles to follow than those of experimental physics. In our +<span class="pagenum" id="Page_18">[Pg 18]</span> +immediate experience we find bodies variously formed and constituted; +only with such can we make our observations and experiments. Their +actions are made up of the actions which each of their parts +contributes to the sum of the whole; and hence, if we wish to know the +simplest and most general law of the action of masses and substances +found in nature upon one another, and if we wish to divest these +laws of the accidents of form, magnitude, and position of the bodies +concerned, we must go back to the laws 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.</p> + +<p>The characteristic properties of the elementary volumes of different +bodies are to be found experimentally, 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 mathematical +physics only investigates the laws of action of the elements of a body +independently of the accidents of form, in a purely empirical manner, +and is therefore just as much under the control of experience as what +are called experimental physics. In principle they are not at all +<span class="pagenum" id="Page_19">[Pg 19]</span> +different, and the former only continues the function of the latter, +in order to arrive at still simpler and still more general laws of +phenomena.</p> + +<p>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 antagonism. 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 experimentalise profitably +who have a clear-sighted knowledge of theory, and know how to propound +and pursue the right questions; and, on the other hand, only those can +theorise with advantage who have great practice in experiments. The +discovery of spectrum analysis is the most brilliant example within our +recollection of such an interpenetration of theoretical knowledge and +experimental skill.</p> + +<p>I am not aware whether Magnus subsequently expressed 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 +<span class="pagenum" id="Page_20">[Pg 20]</span> +reaction against the misuse of speculation carried too far, must also +admit that in the older mathematical 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 researches 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.</p> + +<p>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 +progress. And if we are all of the same opinion that the task of +science is to find the <i>Laws of Facts</i>, 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 discovered. But just as we +all, like Magnus, are opposed 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 +<span class="pagenum" id="Page_21">[Pg 21]</span> +facts which fit to no rule, and which also try carefully to avoid a +law, or a possible connection between newly discovered facts.</p> + +<p>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 physical theories hitherto held, +which treated of atoms and forces acting at a distance, was even more +pronounced than with Magnus.</p> + +<p>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 experiment, 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 deflected 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 aperture through which they flow, +partly in consequence of the manner in which they flow to it; and how +<span class="pagenum" id="Page_22">[Pg 22]</span> +their resolution into drops is caused by external agitation. He applied +the principle of the stroboscopic disk in observing the phenomena, by +looking at the jet through small slits in a rotating disk. He grouped +the various phenomena with peculiar tact, so that those among them +which are alike were easily seen, and one elucidated the other. And +if a final mechanical explanation is not always attained, yet the +reason for a great number of characteristic features of the individual +phenomena is plain. In this respect many of his researches—I might +specially commend those on the efflux of jets of water—are excellent +models of what Goethe theoretically advanced, and in his physical +labours endeavoured to accomplish, though with only partial success.</p> + +<p>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 declared: ‘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 +<span class="pagenum" id="Page_23">[Pg 23]</span> +portions, and such as had been softened by heat; or those it 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 conductivity for heat, using this expression in its +ordinary meaning, and he had to content himself with the obviously +imperfect explanation that two pieces of the same metal at different +temperatures acted like dissimilar 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.</p> + +<p>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. +<span class="pagenum" id="Page_24">[Pg 24]</span></p> + +<p>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 subjugated the forces of nature to +the dominion of the mind.</p> + +<p>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 physical 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. +<span class="pagenum" id="Page_25">[Pg 25]</span></p> + +<p>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 happiness 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.</p> + +<p>And in conclusion we must add, in so far as thoughtfulness, purity of +intention, moral and intellectual 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 satisfactory 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. +<span class="pagenum"><a id="Page_26">[Pg 26]</a></span></p> + +<hr class="chap x-ebookmaker-drop"> +<div class="chapter"> +<p><span class="pagenum"><a id="Page_27">[Pg 27]</a></span></p> + +<h2 class="nobreak"> ON THE ORIGIN AND SIGNIFICANCE<br>OF GEOMETRICAL AXIOMS.</h2> +</div> + +<p class="center"><i>Lecture delivered in the Docenten Verein<br> in Heidelberg, in the year +1870.</i></p> + +<p>The fact that a science can exist and can be developed 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 +inconsistency have so little dared to raise their eyes. It escapes the +tedious and troublesome task of collecting experimental facts, which is +the province of the natural sciences in the strict sense of the word; +<span class="pagenum" id="Page_28">[Pg 28]</span> +the sole form of its scientific method is deduction. Conclusion 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 architecture, +the construction of machinery no less than mathematical physics, are +continually calculating relations of space of the most varied kind +by geometrical principles; they expect that the success of their +constructions and experiments shall agree with these calculations; and +no case is known in which this expectation has been falsified, provided +the calculations were made correctly and with sufficient data.</p> + +<p>Indeed, the fact that geometry exists, and is capable 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 +philosophical systems, that there can be a cognition of principles +destitute of any bases drawn from experience. In the answer to Kant’s +celebrated question, ‘How are synthetical principles <i>a priori</i> +possible?’ geometrical axioms are certainly those examples which appear +to show most decisively that synthetical principles are <i>a priori</i> +possible at all. The circumstance that such principles exist, and force +themselves on our conviction, is regarded as a proof that space is an +<span class="pagenum" id="Page_29">[Pg 29]</span> +<i>a priori</i> mode of all external perception. It appears thereby to +postulate, for this <i>a priori</i> 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 peculiarities +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.⁠<a id="FNanchor_2_2" href="#Footnote_2_2" class="fnanchor">[2]</a></p> + +<p>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.</p> + +<p>I intend to give you an account of a series of recent and closely +connected mathematical researches which are concerned with the +<span class="pagenum" id="Page_30">[Pg 30]</span> +geometrical axioms, their relations to experience, with the question +whether it is logically possible to replace them by others.</p> + +<p>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 +virtually 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.</p> + +<p>Anyone who has entered the gates of the first elementary 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.</p> + +<p>For example, the proposition that if the shortest line drawn between +two points is called a <i>straight</i> line, there can be only one such +<span class="pagenum" id="Page_31">[Pg 31]</span> +straight line. Again, it is an axiom that through any three points +in space, not lying in a straight line, a plane may be drawn, i.e. +a surface which will wholly include every straight line joining any +two of its points. Another axiom, about which there has been much +discussion, affirms that through a point lying without a straight +line only one straight line can be drawn parallel to the first; two +straight lines that lie in the same plane and never meet, however far +they may be produced, being called parallel. There are also axioms that +determine the number of dimensions of space and its surfaces, lines and +points, showing how they are continuous; as in the propositions, that +a solid is bounded by a surface, a surface by a line and a line by a +point, that the point is indivisible, that by the movement of a point +a line is described, by that of a line a line or a surface, by that of +a surface a surface or a solid, but by the movement of a solid a solid +and nothing else is described.</p> + +<p>Now what is the origin of such propositions, unquestionably 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 +<span class="pagenum" id="Page_32">[Pg 32]</span> +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 entangled 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.</p> + +<p>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 prescribed for the demonstration we +have not involuntarily 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 +<span class="pagenum" id="Page_33">[Pg 33]</span> +known that problems of construction play an essential 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 establish the existence of definite figures. They show that +points, straight lines, or circles such as the problem requires to be +constructed are possible under all conditions, 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 supposed 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 translation 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.</p> + +<p>I offer these remarks, at first only to show what difficulties attend +<span class="pagenum" id="Page_34">[Pg 34]</span> +the complete analysis of the pre-suppositions we make, in employing +the common constructive 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 submitted to it that is not already contained in the +equations 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.</p> + +<p>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 +<span class="pagenum" id="Page_35">[Pg 35]</span> +would be generated by a solid moving out of the space we know. By +the much-abused expression ‘to represent’ or ‘to be able to think +how something happens’ I understand—and I do not see how anything +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.</p> + +<p>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 <i>geodetic lines</i> of +the surface on which they live; lines such as are described by a +<i>tense</i> thread laid along the surface, and which can slide upon it +freely. I will henceforth speak of such lines as the <i>straightest</i> +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 of my +<span class="pagenum" id="Page_36">[Pg 36]</span> +non-mathematical hearers without giving rise to misconception.</p> + +<p>Now if beings of this kind lived on an infinite plane, their geometry +would be exactly the same as our planimetry. They would affirm +that only one straight line is possible between two points; that +through a third point lying without this line only one line can be +drawn parallel to it; that the ends of a straight line never meet +though it is produced to infinity, and so on. Their space might be +infinitely extended, but even if there were limits to their movement +and perception, they would be able to represent to themselves a +continuation beyond these limits; and thus their space would appear +to them infinitely extended, 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.</p> + +<p>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 between the two points, but also the other +or larger arc of the same great circle is a geodetic or straightest +line, <i>i.e.</i> every smaller part of it is the shortest line between +<span class="pagenum" id="Page_37">[Pg 37]</span> +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. +Accordingly, 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.</p> + +<p>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.</p> + +<p>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 of +<span class="pagenum" id="Page_38">[Pg 38]</span> +all were the same; nor are more examples necessary to show that +geometrical axioms must vary according to the kind of space inhabited +by beings whose powers of reason are quite in conformity with ours. But +let us proceed still farther.</p> + +<p>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.</p> + +<p>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 +<span class="pagenum" id="Page_39">[Pg 39]</span> +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 curvature of surfaces.⁠<a id="FNanchor_3_3" href="#Footnote_3_3" class="fnanchor">[3]</a> +The ‘measure of curvature,’ as he called it, <i>i.e.</i> the reciprocal +of the product of the greatest and least radii of curvature, must be +everywhere equal over the whole extent of the surface.</p> + +<p>Gauss showed at the same time that this measure of curvature is not +changed if the surface is bent without 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.</p> + +<p>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 +<span class="pagenum" id="Page_40">[Pg 40]</span> +plane, but in which the axiom of parallels does not hold good. This +is a kind of curved surface which is, as it were, geometrically +the counterpart of a sphere, and which has therefore been called +the <i>pseudospherical surface</i> by the distinguished Italian +mathematician E. Beltrami, who has investigated its properties.⁠<a id="FNanchor_4_4" href="#Footnote_4_4" class="fnanchor">[4]</a> +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 directions, 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.</p> + +<p>Like the plane and the sphere, pseudospherical surfaces have their +measure of curvature constant, so that every piece of them can be +<span class="pagenum" id="Page_41">[Pg 41]</span> +exactly applied to every 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 pseudospherical +surfaces, because the two principal curvatures of a saddle-shaped +surface have their concavity turned opposite ways.</p> + +<p>A strip of a pseudospherical surface may, for example, be +represented by the inner surface (turned towards the axis) of a solid +anchor-ring. If the plane figure <i>aabb</i> (<a href="#FIG_1">Fig. 1</a>) +is made to revolve on its axis of symmetry AB, the two arcs <i>ab</i> +will describe a pseudospherical concave-convex surface like that of the +ring. Above and below, towards <i>aa</i> and <i>bb</i>, 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 pseudospherical surface may be rolled up +into the shape of a champagne-glass (<a href="#FIG_2">Fig. 2</a>), 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 +<span class="pagenum" id="Page_42">[Pg 42]</span> +and drawn along the surface of the ring or glass, can it be brought to +places of different flexure, at which farther continuation of the piece +is possible.</p> + +<p>In this way too the straightest lines of the pseudospherical 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 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.</p> + +<div class="figcontainer"> + <div class="figsub"> + <img id="FIG_1" src="images/i_042a.jpg" alt="" width="250" height="256" > + <p class="f120"><span class="smcap">Fig. 1.</span></p> + </div> + <div class="figsub"> + <img id="FIG_2" src="images/i_042b.jpg" alt="" width="240" height="258" > + <p class="f120"><span class="smcap">Fig. 2.</span></p> + </div> +</div> + +<p>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 +<span class="pagenum" id="Page_43">[Pg 43]</span> +N. J. Lobatchewsky, professor of mathematics at Kasan,⁠<a id="FNanchor_5_5" href="#Footnote_5_5" class="fnanchor">[5]</a> +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.</p> + +<p>Thus we see that in the geometry of two dimensions 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.</p> + +<p>The difference between plane and spherical geometry 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 infinite +continuation of pseudospherical surfaces. Inhabiting, as we do, a +space of three dimensions and endowed with organs of sense for their +<span class="pagenum" id="Page_44">[Pg 44]</span> +perception, we can represent to ourselves the various cases in which +beings on a surface might have to develop their perception 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 analogous 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.</p> + +<p>There is however another way of treating geometry scientifically. All +known space-relations are measurable, 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 <i>analytical geometry</i>, 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 <i>shortest</i> 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 +<span class="pagenum" id="Page_45">[Pg 45]</span> +intersecting them, are equal; or it may be laid down instead that the +sum of the angles of any triangle is equal to two right angles. These, +also, are determinations of quantity.</p> + +<p>Now we may start with this view of space, according to which the +position of a point may be determined 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. Riemann of Göttingen.⁠<a id="FNanchor_6_6" href="#Footnote_6_6" class="fnanchor">[6]</a> +It has the peculiar advantage that all its operations consist in pure +calculation of quantities, which quite obviates the danger of habitual +perceptions being taken for necessities of thought.</p> + +<p>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. +<span class="pagenum" id="Page_46">[Pg 46]</span> +Riemann calls a system of differences in which one thing can be +determined by <i>n</i> measurements an ‘<i>n</i>fold extended +aggregate’ or an ‘aggregate of <i>n</i> 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 dimensions, +inasmuch as each colour, according to the investigations of Thomas +Young and of Clerk Maxwell,⁠<a id="FNanchor_7_7" href="#Footnote_7_7" class="fnanchor">[7]</a> +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.</p> + +<p>In the same way we may consider the system of simple tones⁠<a id="FNanchor_8_8" href="#Footnote_8_8" class="fnanchor">[8]</a> +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 +between space of three dimensions and other aggregates. We can, as +we know from daily experience, compare the vertical distance of two +points with the horizontal distance of two others, because we can apply +a measure first to the one pair and then to the other. But we cannot +compare the difference between two tones of equal pitch and different +intensity, with that between two tones of equal intensity and different +pitch. Riemann showed, by considerations of this kind, that the +<span class="pagenum" id="Page_47">[Pg 47]</span> +essential foundation 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.⁠<a id="FNanchor_9_9" href="#Footnote_9_9" class="fnanchor">[9]</a> +Then he showed that the kind of free mobility without change of form +which belongs to bodies in our space can only exist when certain +quantities yielded by the calculation⁠<a id="FNanchor_10_10" href="#Footnote_10_10" class="fnanchor">[10]</a>—quantities +that coincide with Gauss’s measure of surface-curvature when they are +expressed for surfaces—have everywhere an equal value. For this reason +Riemann 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,⁠<a id="FNanchor_11_11" href="#Footnote_11_11" class="fnanchor">[11]</a> +I will once more observe that this so-called measure of space-curvature +is a quantity obtained by purely analytical calculation, and that +its introduction involves no suggestion of relations that would have +a meaning only for sense-perception. The name is merely taken, as a +short expression for a complex relation, from the one case in which the +quantity designated admits of sensible representation.</p> + +<p><span class="pagenum" id="Page_48">[Pg 48]</span> +Now whenever the value of this measure of curvature in any space is +everywhere zero, that space everywhere conforms to the axioms of +Euclid; and it may be called a <i>flat</i> (<i>homaloid</i>) space in +contradistinction to other spaces, analytically constructible, that +may be called <i>curved</i>, 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.</p> + +<p>If the measure of curvature is positive we have <i>spherical</i> +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 <i>pseudospherical</i> 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 intersect another given straightest line in that +surface.</p> + +<p>Beltrami⁠<a id="FNanchor_12_12" href="#Footnote_12_12" class="fnanchor">[12]</a> +has rendered these last relations imaginable by showing that the +points, lines, and surfaces of a pseudospherical space of +<span class="pagenum" id="Page_49">[Pg 49]</span> +three dimensions, can be so portrayed in the interior of a sphere +in Euclid’s homaloid 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 direction 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.</p> + +<p>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 +distinguishable from, zero. This latter definition is given in the +axioms of straight lines and parallels.</p> + +<p>Whilst Riemann entered upon this new field from the side of the most +general and fundamental questions of analytical geometry, I myself +<span class="pagenum" id="Page_50">[Pg 50]</span>arrived at similar +conclusions,⁠<a id="FNanchor_13_13" href="#Footnote_13_13" class="fnanchor">[13]</a> +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. Riemann starts by assuming the +above-mentioned algebraical expression which represents in the most +general form the distance between two infinitely near points, and +deduces therefrom, 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.</p> + +<p>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 +<span class="pagenum" id="Page_51">[Pg 51]</span> +of a point, marks the number of the dimensions 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.</p> + +<p>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 definition can, in the first +instance, run only as follows: Between the co-ordinates of any two +points belonging to a solid body, there must be an equation which, +however 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.</p> + +<p>However indeterminate in appearance, this definition 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</p> + +<ul class="index"> +<li class="isub3">AB, AC, AD, AE,</li> +<li class="isub5">BC, BD, BE,</li> +<li class="isub7">CD, CE,</li> +<li class="isub9">DE,</li> +</ul> + +<p class="no-indent"><span class="pagenum" id="Page_52">[Pg 52]</span> +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-ordinates 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 <i>n</i> +independent equations we can determine <i>n</i> contained quantities, +and if we have more than <i>n</i> equations, the superfluous ones +must be deducible from the first <i>n</i>. 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 dimensions 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 assumption for the +definition of solidity, becomes quite sufficient to determine the kind +of equations holding between the co-ordinates of two points rigidly +connected.</p> + +<p>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 something that need not be. When in our space of three +<span class="pagenum" id="Page_53">[Pg 53]</span> +dimensions two points of a solid body are kept fixed, its movements +are limited to rotations round the straight line connecting them. +If we turn it completely round once, it again occupies exactly the +position it had at first. This fact, that rotation in one direction +always brings a solid body back into its original 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 increased +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.</p> + +<p>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 +distinction of different spaces according to their measure of curvature.</p> + +<p>It still remained to be seen whether the laws of motion, as dependent +on moving forces, could also be consistently transferred to spherical +<span class="pagenum" id="Page_54">[Pg 54]</span> +or pseudospherical space. This investigation has been carried out by +Professor Lipschitz of Bonn.⁠<a id="FNanchor_14_14" href="#Footnote_14_14" class="fnanchor">[14]</a> +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.</p> + +<p>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. <i>Necessities of +thought</i>, 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 curvature from that of Euclid’s flat +<span class="pagenum" id="Page_55">[Pg 55]</span> +space. But if we can imagine such spaces of other sorts, it cannot be +maintained that the axioms of geometry are necessary consequences of an +<i>à priori</i> transcendental form of intuition, as Kant thought.</p> + +<p>The distinction between spherical, pseudospherical, and Euclid’s +geometry depends, as was above observed, on the value of a certain +constant called, by Riemann, 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 pseudospherical, 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 +immeasurably distant fixed stars equal to zero (in pseudospherical +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 Riemann 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. +<span class="pagenum" id="Page_56">[Pg 56]</span></p> + +<p>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 theodolite 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 geometrical 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 temperature, or from gravity acting +differently at different places.</p> + +<p>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 measuring arcs and angles in the visual field.</p> + +<p>Every comparative estimate of magnitudes or measurement of their +<span class="pagenum" id="Page_57">[Pg 57]</span> +spatial relations proceeds therefore upon a supposition as to the +behaviour of certain physical things, either the human body or other +instruments 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.</p> + +<p>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 increased 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 irregularities. A well-made convex mirror of moderate +<span class="pagenum" id="Page_58">[Pg 58]</span> +aperture represents the objects in front of it as apparently 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 surface +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 +<span class="pagenum" id="Page_59">[Pg 59]</span> +out upon our world as we can look into theirs, without overstepping the +boundary, they must declare it to be a picture in a spherical mirror, +and would speak of us just as we speak of them; and if two inhabitants +of the different worlds could communicate with one another, neither, +so far as I can see, would be able to convince the other that he had +the true, the other the distorted, relations. Indeed I cannot see that +such a question would have any meaning at all, so long as mechanical +considerations are not mixed up with it.</p> + +<p>Now Beltrami’s representation of pseudospherical space in a sphere of +Euclid’s space, is quite similar, except 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 mathematical expression.⁠<a id="FNanchor_15_15" href="#Footnote_15_15" class="fnanchor">[15]</a> +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. +<span class="pagenum" id="Page_60">[Pg 60]</span></p> + +<p>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 continue 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,⁠<a id="FNanchor_16_16" href="#Footnote_16_16" class="fnanchor">[16]</a> +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 +<span class="pagenum" id="Page_61">[Pg 61]</span> +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.</p> + +<p>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 +prismatically to resemble pieces of one continuous larger lens. With +these, like the convex mirror, we see remote objects 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 +hundred 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 pseudospherical space. In short, pseudospherical space +<span class="pagenum" id="Page_62">[Pg 62]</span> +would not seem to us very strange, comparatively speaking; we should +only at first be subject to illusions in measuring by eye the size and +distance of the more remote objects.</p> + +<p>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 prismatic +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 perspective picture. +<span class="pagenum" id="Page_63">[Pg 63]</span></p> + +<p>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 contraction 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 coherence being of course +maintained only in as far as their elasticity permitted their bending +without breaking. 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.</p> + +<p>These remarks will suffice to show the way in which we can infer +from the known laws of our sensible 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 represent to ourselves the look of +a pseudospherical world in all directions just as we can develop the +conception of it. Therefore it cannot be allowed that the axioms of our +<span class="pagenum" id="Page_64">[Pg 64]</span> +geometry depend on the native form of our perceptive faculty, or are in +any way connected with it.</p> + +<p>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 perfectly +new, we find ourselves by reason of our bodily organisation quite +unable to represent a fourth dimension.</p> + +<p>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 compare, divide, and +measure them. All space-measurements, and therefore in general all +ideas of quantities applied to space, assume the possibility of figures +moving 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 +<span class="pagenum" id="Page_65">[Pg 65]</span> +any time or place, and after any revolution. We cannot, however, decide +by pure geometry, and without mechanical considerations, whether the +coinciding bodies may not both have varied in the same sense.</p> + +<p>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 distensions 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 +velocity in a straight line, is not adapted to the image of the world +in the convex mirror. The path would indeed be straight, but the +velocity would depend upon the place.</p> + +<p>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 +<span class="pagenum" id="Page_66">[Pg 66]</span> +therewith, any more than natural bodies do ever in fact correspond +exactly to the abstract notion we have obtained of them by induction. +Taking the notion of rigidity thus as a mere ideal, a strict Kantian +might certainly look upon the geometrical axioms as propositions given, +<i>à priori</i>, 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 +qualities and deportment a body must have to be recognised as rigid.</p> + +<p>But if to the geometrical axioms we add propositions relating to +the mechanical properties of natural bodies, were it only the axiom +of inertia, or the single proposition, that the mechanical and +physical properties 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 +experience. 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 elastic solids of great resistance, retain the same +<span class="pagenum" id="Page_67">[Pg 67]</span> +form in every part of space if no external force affects them, is a +single case falling under the general principle.</p> + +<p>In conclusion, I do not, of course, maintain that mankind 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 impossible, of every geometrical +representation at variance with this fact. For this no knowledge +of the necessary 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 numerous and attentive observations—an intuition of +the kind the artist possesses of the objects he is to represent, 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 <i>intuition</i> 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 +<span class="pagenum" id="Page_68">[Pg 68]</span> +enough been taken by metaphysicians for <i>à priori</i> principles, is +a point on which I need not insist.</p> + +<hr class="tb"> + +<p>To sum up, the final outcome of the whole inquiry may be thus +expressed:—</p> + +<p>(1.) The axioms of geometry, taken by themselves out of all connection +with mechanical propositions, represent no relations of real things. +When thus isolated, 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 geometry.</p> + +<p>(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 +observations, 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. +<span class="pagenum" id="Page_69">[Pg 69]</span></p> + +<h3>APPENDIX.</h3> + +<p>The elements of the geometry of spherical space are most easily obtained +by putting for space of four dimensions the equation for the sphere</p> + +<p class="f110"><i>x</i>² + <i>y</i>² +<i>z</i>² + <i>t</i>² = <i>R</i>²<span class="ws3">(1.)</span></p> + +<p class="no-indent">and for the distance <i>ds</i> between the points</p> + +<p class="f110 no-wrap">(<i>x</i>, <i>y</i>, <i>z</i>, <i>t</i>) and +<span class="fs_150">[</span>(<i>x</i> + <i>dx</i>) (<i>y</i> + <i>dy</i>) +(<i>z</i> + <i>dz</i>) (<i>t</i> + <i>dt</i>)<span class="fs_150">]</span></p> + +<p class="f110"> the value <i>ds</i>² = <i>dx</i>² + <i>dy</i>² + <i>dz</i>² + <i>dt</i>²<span class="ws3">(2.)</span></p> + +<p class="no-indent">It is easily found by means of the methods used for three dimensions +that the shortest lines are given by equations of the form</p> + +<table class="spb1 fs_110"> + <tbody><tr> + <td class="tdl"><i>ax</i> + <i>by</i> + <i>cz</i> + <i>ft</i> = 0</td> + <td class="tdc_wsp" rowspan="2"><img src="images/cbr-3.jpg" alt="" width="16" height="57" ></td> + <td class="tdr" rowspan="2"><span class="ws3">(3.)</span></td> + </tr><tr> + <td class="tdl"><i>αx</i> + <i>βy</i> + <i>γz</i> + <i>φt</i> = 0</td> + </tr> + </tbody> +</table> + +<p class="no-indent">in which <i>a</i>, <i>b</i>, <i>c</i>, <i>f</i>, +as well as <i>α</i>, <i>β</i>, <i>γ</i>, <i>φ</i>, are constants.</p> + +<p>The length of the shortest arc, <i>s</i>, between the points (<i>x</i>, +<i>y</i>, <i>z</i>, <i>t</i>), and (<i>ξ</i>, <i>η</i>, <i>ζ</i>, +<i>τ</i>) follows, as in the sphere, from the equation</p> + +<table class="spb1 fs_120"> + <tbody><tr> + <td class="tdl" rowspan="2">cos </td> + <td class="tdc_wsp bb"><i>s</i></td> + <td class="tdc_wsp" rowspan="2">=</td> + <td class="tdl bb"><i>xξ</i> + <i>yη</i> + <i>zζ</i> + <i>tτ</i></td> + <td class="tdr" rowspan="2"><span class="ws3">(4.)</span></td> + </tr><tr> + <td class="tdc"><i>R</i></td> + <td class="tdc"><i>R</i>²</td> + </tr> + </tbody> +</table> + +<p>One of the co-ordinates may be eliminated from the values given in 2 to +4, by means of <b>equation 1</b>, and the expressions then apply to space of +three dimensions.</p> + +<p>If we take the distances from the points</p> + +<p class="f120"><i>ξ</i> = <i>η</i> = <i>ζ</i> = 0</p> + +<p><span class="pagenum" id="Page_70">[Pg 70]</span></p> + +<p class="no-indent">from which <b>equation 1</b> gives <i>τ</i> = <i>R</i>, then,</p> + +<table class="spb1 fs_120"> + <tbody><tr> + <td class="tdl" rowspan="2">sin </td> + <td class="tdc fs_200" rowspan="2">(</td> + <td class="tdc_wsp bb"><i>s</i>₀</td> + <td class="tdc fs_200" rowspan="2">)</td> + <td class="tdc_wsp" rowspan="2">=</td> + <td class="tdl bb"><i>σ</i></td> + </tr><tr> + <td class="tdc"><i>R</i></td> + <td class="tdc"><i>R</i></td> + </tr> + </tbody> +</table> + +<p class="f110">in which</p> + +<p class="f120"><i>σ</i> = √<span class="over"><i>x</i>² + <i>y</i>² + <i>z</i>²</span></p> + +<p class="f110">or,</p> + +<table class="spb1 fs_120"> + <tbody><tr> + <td class="tdl" rowspan="2"><i>s</i>₀ = <i>R</i> . arc sin </td> + <td class="tdc fs_200" rowspan="2">(</td> + <td class="tdc_wsp bb"><i>σ</i></td> + <td class="tdc fs_200" rowspan="2">)</td> + <td class="tdc_wsp" rowspan="2">=</td> + <td class="tdl" rowspan="2"><i>R</i> . arc tang </td> + <td class="tdc fs_200" rowspan="2">(</td> + <td class="tdc_wsp bb"><i>σ</i></td> + <td class="tdc fs_200" rowspan="2">)</td> + <td class="tdr" rowspan="2"><span class="ws3">(5.)</span></td> + </tr><tr> + <td class="tdc"><i>R</i></td> + <td class="tdc"><i>t</i></td> + </tr> + </tbody> +</table> + +<p class="no-indent">In this, <i>s</i>₀ is the distance of the +point <i>x</i>, <i>y</i>, <i>z</i>, measured from the centre of the +co-ordinates.</p> + +<p>If now we suppose the point <i>x</i>, <i>y</i>, <i>z</i>, of spherical +space, to be projected in a point of plane space whose co-ordinates are +respectively</p> + +<table class="spb1 fs_120"> + <tbody><tr> + <td class="tdl fs_120" rowspan="2">χ</td> + <td class="tdc_wsp" rowspan="2">=</td> + <td class="tdc_wsp bb"><i>Rx</i></td> + <td class="tdl" rowspan="2"><span class="ws2"> </span></td> + <td class="tdl fs_120" rowspan="2">ϒ</td> + <td class="tdc_wsp" rowspan="2">=</td> + <td class="tdc_wsp bb"><i>Ry</i></td> + <td class="tdl" rowspan="2"><span class="ws2"> </span></td> + <td class="tdl fs_120" rowspan="2">ζ</td> + <td class="tdc_wsp" rowspan="2">=</td> + <td class="tdc_wsp bb"><i>Rz</i></td> + </tr><tr> + <td class="tdc"><i>t</i></td> + <td class="tdc"><i>t</i></td> + <td class="tdc"><i>t</i></td> + </tr> + </tbody> +</table> + +<table class="spb1 fs_120"> + <tbody><tr> + <td class="tdl" rowspan="2">χ² + ϒ² + ζ² = </td> + <td class="tdc" rowspan="2"><i>r</i>² = </td> + <td class="tdc_wsp bb"><i>R</i>²<i>σ</i>²</td> + </tr><tr> + <td class="tdc"><i>t</i>²</td> + </tr> + </tbody> +</table> + +<p class="no-indent">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 χ, ϒ, ζ, by straight lines. For very small +values of <i>x</i>, <i>y</i>, <i>z</i>, <i>t</i> = <i>R</i>, and</p> + +<p class="f120">χ = <i>x</i>, ϒ = <i>y</i>, ζ = <i>z</i></p> + +<p class="no-indent">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</p> + +<table class="spb1 fs_120"> + <tbody><tr> + <td class="tdl" rowspan="2"><i>s</i>₀ = <i>R</i> . arc tang </td> + <td class="tdc fs_200" rowspan="2">(</td> + <td class="tdc" rowspan="2">± </td> + <td class="tdc_wsp bb"><i>r</i></td> + <td class="tdc fs_200" rowspan="2">)</td> + <td class="tdr" rowspan="2"><span class="ws3">(6.)</span></td> + </tr><tr> + <td class="tdc"><i>R</i></td> + </tr> + </tbody> +</table> + +<p class="no-indent">In this, <i>r</i> may be infinite; but every point +of plane space must be the projection of two points of the sphere, +one for which <b><i>s</i>₀ < ½<i>R</i>π</b>, one for which <b><i>s</i>₀ > +½<i>R</i>π</b>. The extension in the direction of <i>r</i> is then</p> + +<table class="spb1 fs_120"> + <tbody><tr> + <td class="tdl bb"><i>ds</i>₀</td> + <td class="tdc_wsp" rowspan="2">=</td> + <td class="tdc bb"><i>R</i>²</td> + </tr><tr> + <td class="tdc"><i>dr</i></td> + <td class="tdc"><i>R</i>² + <i>r</i>²</td> + </tr> + </tbody> +</table> + +<p><span class="pagenum" id="Page_71">[Pg 71]</span> +In order to obtain corresponding expressions for pseudospherical space, +let <i>R</i> and <i>t</i> be imaginary; that is,</p> + +<p class="f120"><i>R</i> = ℛ<i>i</i>, and <i>t</i> = τ<i>i</i>.</p> +<p class="center"><b>Equation 6</b> gives then</p> + +<table class="spb1 fs_120"> + <tbody><tr> + <td class="tdc" rowspan="2">tang </td> + <td class="tdc bb"><i>s</i>₀</td> + <td class="tdc_wsp" rowspan="2">=</td> + <td class="tdc" rowspan="2">± </td> + <td class="tdc_wsp bb"><i>r</i></td> + </tr><tr> + <td class="tdc"><i>i</i>ℛ</td> + <td class="tdc"><i>i</i>ℛ</td> + </tr> + </tbody> +</table> + +<p class="no-indent">from which, eliminating the imaginary form, we get</p> + +<table class="spb1 fs_120"> + <tbody><tr> + <td class="tdc" rowspan="2"><i>s</i>₀ = ½ℛ  log. nat. </td> + <td class="tdc bb">ℛ + <i>r</i></td> + </tr><tr> + <td class="tdc">ℛ - <i>r</i></td> + </tr> + </tbody> +</table> + +<p class="no-indent">Here <i>s</i>₀ has real values only as +long as <i>r</i> = R; for <i>r</i> = ℛ the distance <i>s</i>₀ in +pseudospherical space is infinite. The image in plane space is, on the +contrary, contained in the sphere of radius <i>R</i>, and every point +of this sphere forms only one point of the infinite pseudospherical +space. The extension in the direction of <i>r</i> is</p> + +<table class="spb1 fs_120"> + <tbody><tr> + <td class="tdl bb"><i>ds</i>₀</td> + <td class="tdc_wsp" rowspan="2">=</td> + <td class="tdc bb">ℛ²</td> + </tr><tr> + <td class="tdc"><i>dr</i></td> + <td class="tdc">ℛ² - <i>r</i>²</td> + </tr> + </tbody> +</table> + +<p>For linear elements, on the contrary, whose direction is at right +angles to <i>r</i>, and for which <i>t</i> is unchanged, we have in +both cases</p> + +<table class="spb1 fs_120"> + <tbody><tr> + <td class="tdl bb">√<span class="over"><i>dx</i>² + <i>dy</i>² + <i>dz</i>²</span></td> + <td class="tdc_wsp" rowspan="2">=</td> + <td class="tdc_wsp bb"><i>t</i></td> + <td class="tdc_wsp" rowspan="2">=</td> + <td class="tdc_wsp bb">τ</td> + <td class="tdc_wsp" rowspan="2">=</td> + <td class="tdc_wsp bb"><i>σ</i></td> + </tr><tr> + <td class="tdc">√<span class="over"><i>d</i>χ² + <i>d</i>ϒ² + <i>d</i>ζ²</span></td> + <td class="tdc"><i>R</i></td> + <td class="tdc">ℛ</td> + <td class="tdc"><i>r</i></td> + </tr> + </tbody> +</table> + +<table class="spb1 fs_120"> + <tbody><tr> + <td class="tdc_wsp" rowspan="2"><span class="ws4">=</span></td> + <td class="tdc bb">√<span class="over"><i>x</i>² + <i>y</i>² + <i>z</i>²</span></td> + </tr><tr> + <td class="tdc">√<span class="over">χ² + ϒ² + ζ²</span></td> + </tr> + </tbody> +</table> + +<p><span class="pagenum"><a id="Page_72">[Pg 72]</a></span></p> + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum"><a id="Page_73">[Pg 73]</a></span></p> +<h2 class="nobreak">ON THE RELATION OF OPTICS<br> TO PAINTING.</h2> +</div> + +<p class="center"><i>Being the substance of a series of Lectures<br> +delivered in Cologne, Berlin, and Bonn.</i></p> + +<p>I fear that the announcement of my intention to address 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 frequent 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 physiology +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 +<span class="pagenum" id="Page_74">[Pg 74]</span> +compare myself to a traveller who has entered upon them by a steep and +stony mountain path, but who, in doing so, has passed many a stage from +which a good point of view is obtained. If therefore I relate to you +what I consider I have observed, it is with the understanding that I +wish to regard myself as open to instruction by those more experienced +than myself.</p> + +<p>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 condition 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 elementary 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 insight into its action, and into the principle of its +methods, cannot be complete if this physiological element is not taken +<span class="pagenum" id="Page_75">[Pg 75]</span> +into account. Next to music this seems to predominate more particularly +in painting, and this is the reason why I have chosen painting as the +subject of my present lecture.</p> + +<p>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 suppose we have present +the real objects themselves, and not a picture; but in so far that the +artistic representation 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 instructive 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 accurate, and whose +memory for these images is particularly true. That which long tradition +has handed down to the men most gifted in this respect, and that +which they have found by innumerable experiments in the most varied +<span class="pagenum" id="Page_76">[Pg 76]</span> +directions, as regards means and methods of representation, forms a +series of important and significant facts, which the physiologist, who +has here to learn from the artist, cannot afford to neglect. 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 determining 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.</p> + +<p>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.</p> + +<p>We have not here to do with a discussion of the ultimate objects +and aims of art, but only with an examination 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.</p> + +<p>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 +<span class="pagenum" id="Page_77">[Pg 77]</span> +instructions according to which the artist is to work. I consider it a +mistake to suppose that any kind of æsthetic 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. +<span class="pagenum" id="Page_78">[Pg 78]</span></p> + +<h3>I. <span class="smcap">Form.</span></h3> + +<p>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 idealisation 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 +expression, and a characteristic development, and if possible beautiful +forms, which have perhaps belonged to no living individuals or indeed +any individuals which ever have existed, but only to such a one as +<span class="pagenum" id="Page_79">[Pg 79]</span> +might exist, and as must exist, to produce a vivid perception of any +particular aspect of human existence in its complete and unhindered +development.</p> + +<p>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?</p> + +<p>Since the picture is on a plane surface, this faithful 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 photographer, 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.</p> + +<p>But apart from the fact that any movement of the observer, whereby his +<span class="pagenum" id="Page_80">[Pg 80]</span> +eye changes its position, will produce displacements of the visual +image, different when he stands before objects from those when he +stands before the image, I could speak of only <i>one</i> eye for +which equality of impression is to be established. We however see the +world with <i>two</i> 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 important 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.</p> + +<p>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 different perspective images. With the right eye we +see somewhat more of the right side of objects before us, and also +<span class="pagenum" id="Page_81">[Pg 81]</span> +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 background 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 corresponding 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 different +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 +<span class="pagenum" id="Page_82">[Pg 82]</span> +or riding, the nearer objects are apparently displaced 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 binocular vision and on the movement +of the body are less operative, because in very distant objects the +differences between the images of the two eyes, or between 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 +powerfully and more distinctly on our perception.</p> + +<p>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 +<span class="pagenum" id="Page_83">[Pg 83]</span> +other hand, what are called geometrical projections, 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.</p> + +<p>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, selection, and mode of illumination +of the objects represented. 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 +<span class="pagenum" id="Page_84">[Pg 84]</span> +expression, which impresses itself without fatigue on the observer.</p> + +<p>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.</p> + +<p>The subordinate methods of expressing depth which have been referred +to, depend in the first place on perspective. Nearer objects partially +conceal more distant ones, but can never themselves be concealed by +the latter. If therefore the painter skilfully groups his objects, 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 counteracts 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 +<span class="pagenum" id="Page_85">[Pg 85]</span> +symmetrical halves. The best perspective 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.</p> + +<p>When human habitations are seen in a picture, they represent to the +observer the direction of the horizontal 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 represent.</p> + +<p>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 foreground, and thus their apparent +magnitude furnishes a measure of the distance at which they are placed.</p> + +<p>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 +<span class="pagenum" id="Page_86">[Pg 86]</span> +task to imitate the fine gradation and transitions of light and shade +on rounded surfaces, which are his chief means of expressing 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, produce the impression of reliefs which are +only illuminated 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 completely without exception, that even in stereoscopic +views a falsely placed double shadow may destroy or confuse the entire +illusion.</p> + +<p>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; +<span class="pagenum" id="Page_87">[Pg 87]</span> +and over surfaces which like those of plane or hilly land only present +slightly moving figures, we require light which is almost in the +direction of the surface itself, for only such a one gives shadows. +This is one of the reasons which makes illumination by the rising or +the setting sun so effective. The forms of the landscape become more +distinct. To this must also be added the influence of colour, and of +aerial light, which we shall subsequently discuss.</p> + +<p>Direct illumination from the sun, or from a flame, makes the shadows +sharply defined, and hard. Illumination 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.</p> + +<p>Of more importance for the representation of depth than the elements +hitherto enumerated, and which are more or less of local and accidental +significance, is what is called <i>aerial perspective</i>. By this +we understand the optical action of the light, which the illuminated +masses of air, between the observer and distant objects, give. This +<span class="pagenum" id="Page_88">[Pg 88]</span> +arises from a fine opacity in the atmosphere, which never entirely +disappears. If, in a transparent medium, there are fine transparent +particles of varying density and varying refrangibility, in so far +as they are struck by it, they deflect the light passing through +such a medium, partly by reflection and partly by refraction; to use +an optical expression, they <i>scatter</i> 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 +distributed 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.</p> + +<p>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 +<span class="pagenum" id="Page_89">[Pg 89]</span> +directly below the path of these rays, the air rising from the flame +stands out quite dark in the surrounding bright turbidity; that is to +say, the air rising from the flame has been quite freed from dust. +In the open air, besides dust and occasional smoke, we must often +also take into account the turbidity arising from incipient aqueous +deposits, where the temperature 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.</p> + +<p>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, +<span class="pagenum" id="Page_90">[Pg 90]</span> +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 manner it is bluer and darker when we ascend +high mountains, 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 +<span class="pagenum" id="Page_91">[Pg 91]</span> +latter it is mixed with the light from objects behind; and moreover +it belongs to the coarser turbidity of the lower regions of the +atmosphere, so that it is whiter. In hot countries, 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.</p> + +<p>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.</p> + +<p>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 yellowish 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, especially snow-mountains, appear coloured. +<span class="pagenum" id="Page_92">[Pg 92]</span></p> + +<p>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 +observed, 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, separate 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. Goethe called +attention to the universality of this phenomenon, and endeavoured to +base upon it his theory of colour.</p> + +<p>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.</p> + +<p>This latter is decidedly better for the landscape painter, and the high +<span class="pagenum" id="Page_93">[Pg 93]</span> +transparent landscapes of mountainous 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 magnitudes of +what is seen to stand out, but they are on the other hand favourable to +the artistic unity of colouration.</p> + +<p>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. +<span class="pagenum" id="Page_94">[Pg 94]</span></p> + +<h3>II. <span class="smcap">Shade.</span></h3> + +<p>The circumstances which we have hitherto discussed 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.</p> + +<p>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 +<span class="pagenum" id="Page_95">[Pg 95]</span> +definite, fixed point of view. It might seem to be an object of +pictorial skill to aim at producing, under the given limitations, the +<i>same</i> effect as is produced by the object itself.</p> + +<p>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 overcome.</p> + +<p>Let us begin with the simplest case; with the quantitative 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 picture-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, +<span class="pagenum" id="Page_96">[Pg 96]</span> +which is but slightly altered by admixtures; 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.</p> + +<p>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 intensities 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.</p> + +<p>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 Lambert’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 diameter. 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 +<span class="pagenum" id="Page_97">[Pg 97]</span> +hundred-thousandth part of the brightness of the sun’s disk. The moon +however is a gray body, whose mean brightness is only about one fifth +of that of the purest white.</p> + +<p>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.</p> + +<p>Now pictures which hang in a room are not lighted by the direct light +of the sun, but by that which is reflected 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 1-20th of the brightness of +white directly lighted by the sun; it will generally be only 1-40th, or +even less.</p> + +<p>Hence the painter of the desert, even if he gives up the representation +of the sun’s disk, which is always very imperfect, will have to +represent the glaringly lighted garments of his Bedouins with a white +which, in the most favourable case, shows only the 1-20th 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 +<span class="pagenum" id="Page_98">[Pg 98]</span> +lamp-black, lighted by the sun, is not less than half as bright, as +shaded white in the brighter part of a room.</p> + +<p>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.</p> + +<p>On the other hand, the darkest black which the artist could apply would +be scarcely sufficient to represent 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 ¹/₁₀₀ that of white paper. The brightest colours of a +painter are only about one hundred times as bright as his darkest +shades.</p> + +<p>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 +<span class="pagenum" id="Page_99">[Pg 99]</span> +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.</p> + +<p>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?</p> + +<p>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 +<span class="pagenum" id="Page_100">[Pg 100]</span> +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 insensitive to new impressions, so that it +appreciates strong ones only moderately, and weak ones not at all.</p> + +<p>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 moonlight 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 certain 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 +<span class="pagenum" id="Page_101">[Pg 101]</span> +mere transcript 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.</p> + +<p>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 <i>psycho-physical</i> law of the relations of the various +sensuous impressions to the irritations which produce them. This law +may be expressed as follows: <i>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.</i> Thus, for instance, differences in +intensity of one hundredth of the total amount can be recognised +without great trouble with very different strengths of light, without +exhibiting material differences in the certainty and facility of the estimate, +whether the brightest daylight or the light of a good candle be used. +<span class="pagenum" id="Page_102">[Pg 102]</span></p> + +<div class="figcontainer"> + <div class="figsub"> + <img id="FIG_3" src="images/i_102a.jpg" alt="" width="250" height="269" > + <p class="f120"><span class="smcap">Fig. 3.</span></p> + </div> + <div class="figsub"> + <img id="FIG_4" src="images/i_102b.jpg" alt="" width="240" height="257" > + <p class="f120"><span class="smcap">Fig. 4.</span></p> + </div> +</div> + +<p>The easiest method of producing accurately measurable differences in +the brightness of two white surfaces, depends on the use of rapidly +rotating disks. If a disk, like the adjacent one in <a href="#FIG_3">Fig. 3</a>, +is made to rotate very rapidly (that is, 20 to 30 times in a second), it appears +to the eye to be covered with three grey rings as in <a href="#FIG_4">Fig. 4</a>. +The reader must, however, figure to himself the grey of these rings, as +it appears on the rotating disk of <a href="#FIG_3">Fig. 3</a>, 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 compared 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 <a href="#FIG_3">Fig. 3</a>, 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 +<span class="pagenum" id="Page_103">[Pg 103]</span> +brightness may be obtained, and by this method, when the strength +of the illumination varies, the brightness always diminishes by +the <i>same proportion</i> of its total value. Now it is found, in +accordance with Fechner’s law, that the distinctness of the rings is +nearly constant for very different strengths of light. We exclude, 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.</p> + +<p>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 shadow, +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 comparison 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, +<span class="pagenum" id="Page_104">[Pg 104]</span> +the equally great difference between daylight, on the one hand, and +daylight plus candle-light on the other, can be no longer recognised.</p> + +<p>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 reflects a small fraction, of incident light. For different +intensities of illumination, the difference of brightness between +the two will always correspond to the same fraction 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, notwithstanding the varying strength of light +in the gallery, provided he gives to his colours the same <i>ratio</i> +of brightness as that which actually exists.</p> + +<p>For, in fact, in looking at natural objects, the absolute 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 brightness is for us the perception, from +<span class="pagenum" id="Page_105">[Pg 105]</span> +which we form our 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 nature, to evoke in us the same +conception as to the nature of the bodies seen. A truthful imitation in +this respect would be attained within the limits in which Fechner’s law +holds, if the artist reproduced the fully lighted parts of the objects +which he has to represent with pigments, which, with the same light, +were equal to the colours to be represented. This is approximately the +case. On the whole, the painter chooses coloured pigments which almost +exactly reproduce the colours of the bodies represented, especially +for objects of no great depth, such as portraits, and which are only +darker in the shaded parts. Children begin to paint on this principle; +they imitate one colour by another; and, in like manner also, nations +in which painting has remained in a childish stage. Perfect artistic +painting is only reached when we have succeeded in imitating the action +of light upon the eye, and not merely the pigments; and only when we +look at the object of pictorial representation from this point of view, +will it be possible to understand the variations from nature which +artists have to make in the choice of their scale of colour and of +shade.</p> + +<p>These are, in the first case, due to the circumstance that Fechner’s +<span class="pagenum" id="Page_106">[Pg 106]</span> +law only holds for mean degrees of brightness; while, for a brightness +which is too high or too low, appreciable divergences are met with.</p> + +<p>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 external excitations; the nerves are too soon tired. Very +bright objects appear almost always to be equally bright, even when +there are, in fact, material differences 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.</p> + +<p>It follows from this, that, with moderate illumination, darker objects +become more like the darkest objects, while with greater illumination +brighter objects 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 +<span class="pagenum" id="Page_107">[Pg 107]</span> +characteristic difference between the impression of very powerful and +very feeble illumination.</p> + +<p>When painters wish to represent glowing sunshine, they make all objects +almost equally bright, and thus produce with their moderately bright +colours the impression which the sun’s glow makes upon the dazzled +eye of the observer. If, on the contrary, they wish to represent +moonshine, they only indicate the very brightest objects, particularly +the reflection of moonlight 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 <i>insensitiveness</i> of the eye for differences of +too bright or too feeble lights. If they could employ the colour of +the dazzling brightness 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 impression 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, +<span class="pagenum" id="Page_108">[Pg 108]</span> +for which Fechner’s law holds; and therewith objects are to be +represented whose brightness is beyond the limits of this law.</p> + +<p>We find that the older masters, and pre-eminently Rembrandt, employ the +same deviation, which corresponds 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 aërial 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 remarkable in this shading, and yet these pictures +give particularly bright and vivid aspects of the objects. Hence +they are of particular interest for understanding the principles of +pictorial illumination.</p> + +<p>In order to explain these actions we must, I think, consider that while +Fechner’s law is approximately correct for those mean lights which are +agreeable to the eye, the deviations which are so marked, for too high +<span class="pagenum" id="Page_109">[Pg 109]</span> +or too 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 produced 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 +¹/₁₅₀ or ¹/₁₈₀ of the total intensity can be recognised. The light in +which pictures are looked at is, on the contrary, much feebler; and +if we are to retain the same distinctness of the finest shadows and +of the modelling of the contours which it produces, the gradations of +shade in the picture must be somewhat stronger than corresponds to the +exact luminous intensities. The darkest objects of the picture thereby +become unnaturally 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 gradations are imitated with expression. +<span class="pagenum" id="Page_110">[Pg 110]</span></p> + +<h3>III. <span class="smcap">Colour.</span></h3> + +<p>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 admixtures of +three simple sensations; namely, of red, green, and violet,⁠<a id="FNanchor_17_17" href="#Footnote_17_17" class="fnanchor">[17]</a> +which, by a not improbable supposition of Thomas Young, can be apprehended +quite independently of each other by three different systems of +nerve-fibres. To this independence of the different sensations of +colour corresponds their independence in the gradation of intensity. +Recent measurements⁠<a id="FNanchor_18_18" href="#Footnote_18_18" class="fnanchor">[18]</a> +have shown that the sensitiveness of our eye for feeble shadows is +greatest in the blue and least in the red. A difference of ¹/₂₀₅ to +¹/₂₆₈ of the intensity can be observed in the blue, and with an untired +eye of ¹/₁₆ in the red; or when the colour is dimmed by being looked at +for a long time, a difference of ¹/₅₀ to ¹/₇₀.</p> + +<p><span class="pagenum" id="Page_111">[Pg 111]</span> +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, 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.</p> + +<p>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 +<span class="pagenum" id="Page_112">[Pg 112]</span> +therefore yellowish, and dull white appears bluish. In our ordinary way +of looking at the objects about us, we are not so readily conscious of +this; for the direct comparison of colours of very different shade is +difficult, 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.</p> + +<p>If however to the painter is put the problem of imitating, 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 moonlight, 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 grey. +<span class="pagenum" id="Page_113">[Pg 113]</span></p> + +<p>I will again remind you that these changes of 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.</p> + +<p>The change of colour, like the scale of shade, previously discussed, is +a subjective action which the artist must represent objectively on his +canvas, since moderately bright colours cannot produce them.</p> + +<p>We observe something quite similar in regard to the phenomena of +<i>Contrast</i>. By this term we understand 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.</p> + +<p>The phenomena of contrast are very various, and depend on different +causes. One class, <i>Chevreul’s simultaneous Contrast</i>, 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 +<span class="pagenum" id="Page_114">[Pg 114]</span> +are here met with are often very striking; I will not, however, enter +upon them, for they produce no divergence between the picture and +reality.</p> + +<p>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 impression of each colour changes, +for it is depicted on portions 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 movements of the eye, and has been called by +Chevreul, ‘<i>successive Contrast</i>.’</p> + +<p>We have already seen that the retina is more sensitive 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 perceived. 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.</p> + +<p>I must here remark that the fatigue of the retina by light does not +necessarily extend to the whole surface; but when only a small portion +of this membrane is struck by a minute, defined picture it can also be +locally developed in this part only. +<span class="pagenum" id="Page_115">[Pg 115]</span></p> + +<p>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 <i>negative after-images</i> +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 +surface, 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 impressions of light. A longer time is required in order +to develop 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 +<span class="pagenum" id="Page_116">[Pg 116]</span> +producing sharp photographic portraits, the 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.</p> + +<p>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.⁠<a id="FNanchor_19_19" href="#Footnote_19_19" class="fnanchor">[19]</a> +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,⁠<a id="FNanchor_20_20" href="#Footnote_20_20" class="fnanchor">[20]</a> +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. 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.</p> + +<p><span class="pagenum" id="Page_117">[Pg 117]</span> +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 shifting 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.</p> + +<p>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.</p> + +<p>This also is a phenomenon which is produced more strongly by bright +light and brilliant saturated colours than by fainter light and duller +<span class="pagenum" id="Page_118">[Pg 118]</span> +colours. The artist however, works for the most part with the latter. +He produces most of his tints by mixture; each mixed pigment is, +however, greyer and duller than the pure colour of which it is +mixed, and even the few pigments 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 contrasts, 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 +themselves 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.</p> + +<p>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 +<span class="pagenum" id="Page_119">[Pg 119]</span> +the edge where there is a background of blue, and a rose-red tint where +they impinge 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 predominant 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 +represented by the painter, since the colours of his pictures are not +bright enough to produce the contrast without such help.</p> + +<p>To the series of subjective phenomena, which artists are compelled to +represent objectively in their pictures, must be associated certain +phenomena of <i>irradiation</i>. By this is understood cases in which +any bright 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 +neighbourhood 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 disappears, which covers the whole neighbourhood, and, at the same +<span class="pagenum" id="Page_120">[Pg 120]</span> +time, any objects there may be in the dark part of the field of view +are seen more distinctly. If the flame is partly screened by a ruler, +this appears jagged where the flame projects beyond it. The luminosity +in the neighbourhood of the flame is so intense, 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.</p> + +<p>The cause of this phenomenon is quite similar to that of aërial +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 aërial 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 perfectly 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 crystal. 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.⁠<a id="FNanchor_21_21" href="#Footnote_21_21" class="fnanchor">[21]</a></p> + +<p><span class="pagenum" id="Page_121">[Pg 121]</span> +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.</p> + +<p>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.</p> + +<p>Irradiation also belongs to the subjective phenomena 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.</p> + +<p>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, +<span class="pagenum" id="Page_122">[Pg 122]</span> +but the impression which they have given, or would give, which is to +be imitated, so as to produce as distinct and vivid a conception as +possible 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 <i>gradations</i> 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 +canvas. The fact that they are represented is particularly significant +for the kind of problem which is to be solved by a pictorial +representation.</p> + +<p>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 +<span class="pagenum" id="Page_123">[Pg 123]</span> +them 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 pictorial elements. +<span class="pagenum" id="Page_124">[Pg 124]</span></p> + +<h3>IV. <span class="smcap">Harmony of Colour.</span></h3> + +<p>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 imitating 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 obviating these +evils? Such there are indeed. Frescoes are sometimes viewed in direct +sunlight; transparencies 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 pictures which +<span class="pagenum" id="Page_125">[Pg 125]</span> +took advantage of this. Fresco painting would have led to this; or the +experiments of Münich’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.</p> + +<p>Experiment seems therefore to teach, that moderation of light and of +colours in pictures is ever advantageous, and we need only look at +frescoes in direct sunlight, such as those of the new Pinakothek in +Münich, 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, brilliant +colours were used, even locally and to a moderate extent, which were +intended to represent bright sunlight, 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.</p> + +<p>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 +<span class="pagenum" id="Page_126">[Pg 126]</span> +already seen, to those matters which we can in reality only estimate in +an uncertain manner, such as the absolute intensities of light.</p> + +<p>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 perceiving the outer world, and which admit of the finest +discrimination and observation.</p> + +<p>It has been mentioned above that the discrimination 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 +attention 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 appears 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 construction of living +organisms, whatever their origin, excludes the possibility that in the +majority of healthy individuals an instinct should be developed or +maintain itself which did not serve some definite purpose. +<span class="pagenum" id="Page_127">[Pg 127]</span></p> + +<p>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 +<span class="pagenum" id="Page_128">[Pg 128]</span> +quality—any slight interruption, any tremulousness, any rising or +falling in it—is at once more distinctly recognised by the hearer than +could be the case with a less regular sound; and it seems also that +the powerful excitation which it produces in the ear of the listener, +arouses trains of ideas and passions more strongly than does a feebler +excitation 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 sensitive 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 perception 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 produces 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 +<span class="pagenum" id="Page_129">[Pg 129]</span> +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 expressive for the slightest change of modelling +or of illumination; that is, they are expressive in the artistic sense.</p> + +<p>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, especially 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 +<span class="pagenum" id="Page_130">[Pg 130]</span> +observer. To be able to do this he must be sparing in 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 +one-sided 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 irritation +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 produces 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.</p> + +<p>These relations of the colours to each other have manifestly a great +influence on the degree of pleasure which different combinations of +<span class="pagenum" id="Page_131">[Pg 131]</span> +colours afford. Two colours may, without injury, be juxtaposed, which +indeed are so similar as to look like varieties of the same colour, +produced by varying degrees of light and shade. Thus, upon scarlet the +more shaded parts appear of a carmine, or on a straw-colour they appear +of a golden-yellow.</p> + +<p>If we pass beyond these limits, we arrive at unpleasant 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 +combinations 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 +<span class="pagenum" id="Page_132">[Pg 132]</span> +combination tends towards yellow, and in the former, 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 one-sided fatigue of +the eye, without falling into the baldness of complementary 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 approximately +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 consonance of tones. On the +contrary, a consideration of the facts shows that a number of accessory +influences come into play,⁠<a id="FNanchor_22_22" href="#Footnote_22_22" class="fnanchor">[22]</a> +when once the coloured surface is also to produce, either wholly or in +part, a representation of natural objects or of solid forms, or even if +it only offers a resemblance with the representation of a relief, of +shaded and of non-shaded surfaces. It is moreover often difficult to +establish, as a matter of fact, what are the colours which produce the +<span class="pagenum" id="Page_133">[Pg 133]</span> +harmonic impression. This is pre-eminently the case with pictures +in which the aërial colour, the coloured reflection 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 determination 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 representation, 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.</p> + +<p>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 corresponds 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 +<span class="pagenum" id="Page_134">[Pg 134]</span> +feebly reflected, and is mainly absorbed. Before the shaded parts of +blue bodies, the yellow aërial 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 æsthetic requirements of artistic unity of +composition 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.</p> + +<p>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 aërial 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 +<span class="pagenum" id="Page_135">[Pg 135]</span> +more distinct 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.</p> + +<hr class="tb"> + +<p>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 binocular 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 position +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 aërial light is one of +the most important means of attaining the object.</p> + +<p>We then saw that even the scale of luminous intensity, as met with +in the objects, must be transformed 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. +<span class="pagenum" id="Page_136">[Pg 136]</span></p> + +<p>The artist cannot transcribe Nature; he must translate her; yet this +translation may give us an impression in the highest degree distinct +and forcible, not merely of the objects themselves, but even of the +greatly altered intensities of light under which we view them. The +altered scale is indeed in many cases advantageous, as it gets rid +of everything which, in the actual objects, is too dazzling, and too +fatiguing for the eye. Thus the imitation of Nature in the picture is +at the same time an ennobling of the impression 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 impressions, 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.</p> + +<p>If, in these considerations, my having continually laid much weight on +<span class="pagenum" id="Page_137">[Pg 137]</span> +the lightest, finest, and most accurate sensuous intelligibility of +artistic representation, may seem to many of you as a very subordinate +point—a point which, if mentioned at all by writers on æsthetics, 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.</p> + +<p>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 +corresponding feelings, and direct them towards a common object, +so as to give a vivid perception of all the features 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 something 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 +<span class="pagenum" id="Page_138">[Pg 138]</span> +certainly, 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 comprehend under the name of +intelligibility of the work of art.</p> + +<p>Then the peculiarities of the painters’ technique (<i>Technik</i>), +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.</p> + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_139">[Pg 139]</span></p> +<h2 class="nobreak">ON THE ORIGIN OF THE<br> PLANETARY SYSTEM.</h2> +</div> + +<p class="center"><i>Lecture delivered in Heidelberg and in Cologne, in 1871.</i></p> + +<p>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, +hypotheses, or dreams.</p> + +<p>Of all the subjects to which the thought and imagination 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. +<span class="pagenum" id="Page_140">[Pg 140]</span> +Beneficent and malignant deities, giants, Kronos who devours his +children, Niflheim, with the ice-giant Ymir, who is killed by the +celestial Asas,⁠<a id="FNanchor_23_23" href="#Footnote_23_23" class="fnanchor">[23]</a> +that out of him the world may be constructed—these are all figures +which fill the cosmogonic systems of the more cultivated 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.</p> + +<p>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 <i>General Natural Philosophy and Theory of the +Heavens</i>, 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 +<span class="pagenum" id="Page_141">[Pg 141]</span> +that they belong mostly to 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 originating in some adventitious circumstance; at the +same time the matter they contain is comparatively without originality, +and they are only important from a destructive 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 +<i>Kritik der reinen Vernunft</i> 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 philosophy, 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 +<span class="pagenum" id="Page_142">[Pg 142]</span> +inversion of the historical connection, when Kant’s 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 energetically and more incisively, than +Kant himself if he were still among us.</p> + +<p>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 conclusions as +to the unknown past from known conditions of the present time.</p> + +<p>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, +however, receive direct confirmation, if, in the progress of scientific +<span class="pagenum" id="Page_143">[Pg 143]</span> +knowledge, new facts accrue to those already known, and like them are +explained on the hypothesis; and particularly if survivals of the +processes, assumed to have taken place in the formation of the heavenly +bodies, can be proved to exist in the present.</p> + +<p>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.</p> + +<p>Partly this fact, and partly the fact that the hypothesis in question +has recently been mentioned in popular and scientific books, in +connection with philosophical, ethical, and theological questions, have +emboldened me to speak of it here. I intend not so much to tell you +anything substantially new in reference to it, as to endeavour to give, +as connectedly as possible, the reasons which have led to, and have +confirmed it.</p> + +<p>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 uniformity of natural +<span class="pagenum" id="Page_144">[Pg 144]</span> +laws, our conclusions from present circumstances as to the past, and as +to the future, imperatively lead to an impossible state of things; that +is, to the necessity of an infraction of natural laws, of a beginning +which could not have been due to processes known to us. Hence, to begin +such an investigation as to the possible or probable primeval history +of our present world, is, considered as a question 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.</p> + +<p>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. +<span class="pagenum" id="Page_145">[Pg 145]</span></p> + +<p>In the front rank of all, then, is the law of gravitation. 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.</p> + +<p>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, +<span class="pagenum" id="Page_146">[Pg 146]</span> +and at 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 +instance, 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 yourselves +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 eliminated, in this case, as they are excluded in +the planetary system. +<span class="pagenum" id="Page_147">[Pg 147]</span></p> + +<div class="figcenter"> + <p id="FIG_5" class="f110"><span class="smcap">Fig. 5.</span></p> + <img src="images/i_147.jpg" alt="" width="600" height="393" > +</div> + +<p>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 distances 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 be +circular but elliptical, of the form of the curved line in <a href="#FIG_5">Fig. 5</a>. +But these ellipses lie in both cases differently as regards the attracting +centre. In our model, the attracting force is stronger, the further the +lead sphere is removed from its position of equilibrium. Under these +circumstances, the ellipse of the path has such a position in reference +to the attracting centre, that this is in the centre, <i>c</i>, of the +ellipse. For planets, on the contrary, the attracting force is feebler +<span class="pagenum" id="Page_148">[Pg 148]</span> +the 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, <i>a</i> and <i>b</i>, are two +points which lie symmetrically towards the ends of the ellipse, and +are characterised by the property that the sum of their distances, +<i>am</i> + <i>bm</i>, is the same from any given points.</p> + +<p>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 publishing his important discovery +until it had acquired a direct confirmation; this followed from the +observations, 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.</p> + +<p>In the course of the eighteenth century the power of mathematical +analysis, and the methods of astronomical observation, increased so +far that all the complicated actions, which take place between all the +<span class="pagenum" id="Page_149">[Pg 149]</span> +planets, 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 manifold phenomena which follow therefrom, +was so complete 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.</p> + +<p>But it was not merely in the region of the attraction of our sun that +the law of gravitation was found to hold. With regard to the fixed +<span class="pagenum" id="Page_150">[Pg 150]</span> +stars, it was found that double stars moved about each other in +elliptical paths, and that therefore the same law of gravitation must +hold for them as for our planetary system. The distance of some of +them could be calculated. The nearest of them, α, in the constellation +of the Centaur, is 270,000 times further from the sun than the earth. +Light, which has a velocity of 186,000 miles a second, which traverses +the distance from the sun to the earth in eight minutes, would take +four years to travel from α 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.</p> + +<p>The knowledge of the law of gravitation has here also led to the +discovery of new bodies, as in the case of Neptune. Peters of Altona +found, confirming therein a conjecture of Bessel, that Sirius, the +most brilliant of the fixed stars, moves in an elliptical path about +an invisible centre. This must have been due to an unseen companion, +and when the excellent and powerful telescope of the University of +Cambridge, in the United States, had been set up, this was discovered. +It is not quite dark, but its light is so feeble that it can only be +<span class="pagenum" id="Page_151">[Pg 151]</span> +seen by the most perfect instruments. The mass of Sirius is found to +be 13·76, and that of its satellite 6·71, times the mass of the sun; +their mutual distance is equal to thirty-seven times the radius of the +earth’s orbit, and is therefore somewhat larger than the distance of +Neptune from the sun.</p> + +<p>Another fixed star, Procyon, is in the same case as Sirius, but its +satellite has not yet been discovered.</p> + +<p>You thus see that in gravitation we have discovered 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.</p> + +<p>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 nebulæ.</p> + +<p>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 +<span class="pagenum" id="Page_152">[Pg 152]</span> +decomposed. The formation of the prismatic spectrum depends on the foot +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.</p> + +<p>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.</p> + +<p>The case is different if the light is emitted by an ignited gas, or by +an ignited vapour—that is, a substance vaporised by heat. The spectrum +of such a body consists, then, of one or more, and sometimes even a +great number, of entirely distinct bright lines, whose position and +arrangement in the spectrum is characteristic for the substances of +which the gas or vapour consists, so that it can be ascertained, by +means of spectrum analysis, what is the chemical constitution of the +ignited gaseous body. Gaseous spectra of this kind are shown in the +heavenly space by many nebulæ; for the most part they are spectra which +show the bright line of ignited hydrogen and oxygen, and along with it +<span class="pagenum" id="Page_153">[Pg 153]</span> +a line which, as yet, has never been 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 nebulæ 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.</p> + +<p>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.</p> + +<div class="figcenter"> + <p id="FIG_6" class="f110"><span class="smcap">Fig. 6.</span></p> + <img src="images/i_154a.jpg" alt="" width="600" height="400" > +</div> + +<div class="figcontainer"> + <div class="figsub"> + <p id="FIG_7" class="f120"><span class="smcap">Fig. 7.</span></p> + <img src="images/i_154b.jpg" alt="" width="300" height="284" > + </div> + <div class="figsub"> + <p id="FIG_8" class="f120"><span class="smcap">Fig. 8.</span></p> + <img src="images/i_154c.jpg" alt="" width="175" height="282" > + </div> +</div> + +<p>Far more powerful apparatus was afterwards used by Kirchhoff, and then +by Angström, to push the decomposition of light as far as possible. +<a href="#FIG_6">Fig. 6</a> represents an apparatus with four prisms, constructed by +Steinheil for Kirchhoff. At the further end of the telescope <span class="allsmcap">B</span> +is a screen with a fine slit, representing a fine slice of light, which +can be narrowed or widened by the small screw, and by which the light +<span class="pagenum" id="Page_154">[Pg 154]</span> +under investigation can be allowed to enter. It then passes +through the telescope <span class="allsmcap">B</span>, afterwards +through the four prisms, and finally through the telescope +<span class="allsmcap">A</span>, from which it reaches the eye of the +<span class="pagenum" id="Page_155">[Pg 155]</span> +observer. Figs. <a href="#FIG_7"> 7</a>, <a href="#FIG_8"> 8</a>, +and <a href="#FIG_9"> 9</a> represent small portions of the solar +spectrum as mapped by Kirchhoff, 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 +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.</p> + +<div class="figcenter"> + <p id="FIG_9" class="f110"><span class="smcap">Fig. 9.</span></p> + <img src="images/i_155.jpg" alt="" width="600" height="348" > +</div> + +<p>It follows from this, that the solar atmosphere contains 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, +<span class="pagenum" id="Page_156">[Pg 156]</span> +moreover, how our figs. <a href="#FIG_7"> 7</a>, <a href="#FIG_8"> 8</a>, +and <a href="#FIG_9"> 9</a> 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.</p> + +<p>The spectra of several fixed stars are similarly constituted; 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 α Orionis the +rare metal thallium; and so on.</p> + +<p>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 +<span class="pagenum" id="Page_157">[Pg 157]</span> +especially nitrogen, which constitutes the greater part of our +atmosphere, and hydrogen, an element in water, which indeed is formed +by its combustion. Both have been found in the irresolvable nebulæ, +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.</p> + +<p>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 terrestrial 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 +<span class="pagenum" id="Page_158">[Pg 158]</span> +assumption, that the sun is a dark cool body, surrounded by a +photosphere which only radiates heat and light externally, contains a +physical impossibility.</p> + +<p>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 <i>Chromosphere</i>. +They show the bright lines of hydrogen, of sodium, of magnesium, 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.⁠<a id="FNanchor_24_24" href="#Footnote_24_24" class="fnanchor">[24]</a> +These structures could formerly only be viewed at the time of a +total eclipse of the sun, forming what were called the rose-red +protuberances. 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. +<span class="pagenum" id="Page_159">[Pg 159]</span></p> + +<div class="figcenter"> + <p id="FIG_10" class="f110"><span class="smcap">Fig. 10.</span></p> + <img src="images/i_159.jpg" alt="" width="500" height="476" > +</div> + +<p>On the other hand, there are individual darker parts on the sun’s +surface, what are called <i>sun-spots</i>, 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. <a href="#FIG_10">Fig. 10</a> represents +such a spot according to Padre Secchi, as seen under powerful 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 they are again resolved, +and meanwhile several rotations 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 +<span class="pagenum" id="Page_160">[Pg 160]</span> +very oblique direction. <a href="#FIG_11">Fig. 11</a> represents in +<span class="allsmcap">A</span> to <span class="allsmcap">E</span> +the different aspects of such a spot as it comes near the edge of the sun.</p> + +<div class="figcenter"> + <p id="FIG_11" class="f110"><span class="smcap">Fig. 11.</span></p> + <img src="images/i_160.jpg" alt="" width="600" height="336" > +</div> + +<p>Just on the edge of these spots there are spectroscopic 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 cooler gases from the outer +layers of the sun’s atmosphere 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 +expansion, and partly by radiation into space, must become cooled. At +<span class="pagenum" id="Page_161">[Pg 161]</span> +the same time, they deposit their more difficultly volatile +constituents as fog or cloud. This cooling can only, of course, be +regarded as comparative; 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.</p> + +<p>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 atmosphere, 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.</p> + +<p>We will now pass to the question of the permanence 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, +<span class="pagenum" id="Page_162">[Pg 162]</span> +of the influence of planetary disturbances. By disturbances of the +planetary 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 constant ellipse whose axes would retain the +same direction 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 disturbances due to the +<span class="pagenum" id="Page_163">[Pg 163]</span> +reciprocal attraction of planets upon each other, and on the assumption +that no forces of other kinds have any influence on their motions.</p> + +<p>On our earth we cannot produce such an everlasting 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.</p> + +<p>Hence the fundamental law of mechanics, according 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.</p> + +<p>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 surrounding air in vibration, and +imparts to it some of its own motion. Thus it happens that its velocity +is continually 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 +<span class="pagenum" id="Page_164">[Pg 164]</span> +addition there is the resistance of the air, which resistance is mainly +due to that of the particles of air against each other, due to their +friction against the wheel.</p> + +<p>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, without contact with any body which could produce +friction, and hence their motion seems to be one which never diminishes.</p> + +<p>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?</p> + +<p>From the progress which the knowledge of nature has made since the time +of Laplace, we must now answer both questions in the negative.</p> + +<p>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, +<span class="pagenum" id="Page_165">[Pg 165]</span> +from the size of huge stones to that of dust, are still everywhere +scattered; at any rate, in those parts of space which our earth +traverses.</p> + +<p>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 <i>vis viva</i>, 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 luminiferous ether.⁠<a id="FNanchor_25_25" href="#Footnote_25_25" class="fnanchor">[25]</a></p> + +<p>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 +<span class="pagenum" id="Page_166">[Pg 166]</span> +any diminution to have been perceived 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. +<span class="pagenum" id="Page_167">[Pg 167]</span></p> + +<p>The presence of partly fine and partly coarse heavy masses diffused +in cosmical space is more distinctly 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 atmosphere 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 terrestrial 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 shooting-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 atmosphere, 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 <i>stelle cadente</i>, falling stars, as they +have long been called by poets.</p> + +<p>This enormous velocity with which they enter our atmosphere is +<span class="pagenum" id="Page_168">[Pg 168]</span> +undoubtedly the cause of their becoming 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 proportional to the work expended. If, like the +mechanicians, 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 sufficient +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 +<span class="pagenum" id="Page_169">[Pg 169]</span> +temperatures attainable by terrestrial means scarcely exceed 2,000 +degrees. In fact, the outer crusts of meteoric stones generally show +traces of incipient fusion; and in cases in which observers examined +with sufficient promptitude 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.</p> + +<p>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 calculated 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.</p> + +<p>Many groups are irregularly distributed in space, being probably those +which have already undergone disturbances by planets. There are also +<span class="pagenum" id="Page_170">[Pg 170]</span> +denser swarms which move in regular elliptical orbits, cutting the +earth’s orbit in definite places, and therefore always occur on +particular days of the year. Thus the 10th of August of each year is +remarkable, and every thirty-three years the splendid fireworks of the +12th to the 14th of November repeats itself for a few years. It is +remarkable that certain comets accompany the paths of these swarms, and +give rise to the supposition that the comets gradually split up into +meteoric swarms.</p> + +<p>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 +<span class="pagenum" id="Page_171">[Pg 171]</span> +by the gradual aggregation of fine dust, we see ourselves led to a +primitive condition of fine nebulous masses.</p> + +<p>From this point of view, that the fall of shooting-stars and of +meteorites is perhaps only a small survival of a process which once +built up worlds, it assumes far greater significance.</p> + +<p>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.</p> + +<p>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 considerable exceptions known are the moons of +<span class="pagenum" id="Page_172">[Pg 172]</span> +Uranus, 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, especially 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 +<span class="pagenum" id="Page_173">[Pg 173]</span> +paths would have been as probable as the almost circular 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.</p> + +<p>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 +primitive 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.</p> + +<p>These were the essential features of the considerations 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 extended to the path of the most distant planet, many billions of +cubic miles could contain scarcely a gramme of mass. This ball, when it +<span class="pagenum" id="Page_174">[Pg 174]</span> +had become detached from 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 another 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.</p> + +<p>Our more recent experience as to the nature of star showers teaches us +that this process of the condensation 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. +<span class="pagenum" id="Page_175">[Pg 175]</span></p> + +<p>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.</p> + +<p>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 accumulate in their +structure organic matter, which partly serves the whole animal kingdom +as food, and serves man more particularly as fuel. Coals and lignites, +<span class="pagenum" id="Page_176">[Pg 176]</span> +the sources of power of our steam engines, are remains of primitive +plants—the ancient production of the sun’s rays.</p> + +<p>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.</p> + +<p>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 circulation 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 distantly compared. I have previously +explained the mechanical equivalent of heat. Calculated by that +<span class="pagenum" id="Page_177">[Pg 177]</span> +standard, the work which the sun produces by its radiation is equal to +the constant exertion of 7,000 horse-power for each square foot of the +sun’s surface.</p> + +<p>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 independent of the special +interests of man. These have been generalised and comprehended in the +all-ruling natural law of the <i>Conservation of Force</i>. 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 +consumption. 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, +<span class="pagenum" id="Page_178">[Pg 178]</span> +everlasting and unchangeable like matter itself. It seems as if Goethe +had an idea of this when he makes the earth-spirit speak of himself as +the representative of natural force.</p> + +<ul class="index"> +<li class="isub2">In the currents of life, in the tempests of motion,</li> +<li class="isub2">In the fervour of art, in the fire, in the storm,</li> +<li class="isub6">Hither and thither,</li> +<li class="isub6">Over and under,</li> +<li class="isub6">Wend I and wander.</li> +<li class="isub6">Birth and the grave,</li> +<li class="isub6">Limitless ocean,</li> +<li class="isub6">Where the restless wave</li> +<li class="isub6">Undulates ever</li> +<li class="isub6">Under and over,</li> +<li class="isub6">Their seething strife</li> +<li class="isub6">Heaving and weaving</li> +<li class="isub6">The changes of life.</li> +<li class="isub2">At the whirling loom of time unawed,</li> +<li class="isub2">I work the living mantle of God.</li> +</ul> + +<p>Let us return to the special question which concerns us here: Whence +does the sun derive this enormous store of force which it sends out?</p> + +<p>On earth the processes of combustion are the most abundant source of +heat. Does the sun’s heat originate in a process of this kind? To this +question we can reply with a complete and decided negative, for we +now know that the sun contains the terrestrial elements with which we +are acquainted. Let us select from among them the two, which, for the +smallest mass, produce the greatest amount of heat when they combine; +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 +<span class="pagenum" id="Page_179">[Pg 179]</span> +sun is known, and also the quantity of heat produced by the union of +known weights of oxygen and hydrogen. Calculation shows that under +the above supposition, the heat resulting from their combustion would +be sufficient to keep up the radiation of heat from the sun for 3,021 +years. That, it is true, is a long time, but even profane history +teaches that the sun has lighted and warmed us for 3,000 years, and +geology puts it beyond doubt that this period must be extended to +millions of years.</p> + +<p>Known chemical forces are thus so completely inadequate, 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.</p> + +<p>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 +extraordinarily 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 <i>raised from the earth</i> can drive our clocks, +and that in like manner the gravity of the water rushing down from the +mountains works our mills. +<span class="pagenum" id="Page_180">[Pg 180]</span></p> + +<p>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.</p> + +<p>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 equivalence, previously established, between +heat and mechanical work.</p> + +<p>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 +<span class="pagenum" id="Page_181">[Pg 181]</span> +made up by the fall of meteors, a hypothesis which was propounded by +Mayer, and has been favourably adopted by several other physicists, is +open, according to Sir W. Thomson’s investigations, to objection; 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.</p> + +<p>If, now, there is no present manifestation of force sufficient to cover +the expenditure of the sun’s heat, the sun must originally have had a +store of heat which it gradually gives out. But whence this store? We +know that the cosmical forces alone could have produced it. And here +the hypothesis, previously discussed 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.</p> + +<p>Calculation shows that, assuming the thermal capacity of the sun to be +<span class="pagenum" id="Page_182">[Pg 182]</span> +the same as that of water, the temperature might be raised to +28,000,000 of degrees, if this quantity of heat could ever have been +present in the sun at one time. This cannot be assumed, for such +an increase of temperature would offer the greatest hindrance to +condensation. It is probable rather that a great part of this heat, +which was produced 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.</p> + +<p>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 condense +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.</p> + +<p>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 maintain for an additional +<span class="pagenum" id="Page_183">[Pg 183]</span> +17,000,000 of years the same intensity of sunshine as that which is now +the source of all terrestrial life.</p> + +<p>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 arbitrary assumptions; and, compared with the general uniform +distribution of the internal heat, are somewhat insufficient.</p> + +<p>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 +<span class="pagenum" id="Page_184">[Pg 184]</span> +temperature, and the slower cooling, which characterises larger +masses.⁠<a id="FNanchor_26_26" href="#Footnote_26_26" class="fnanchor">[26]</a> +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.</p> + +<div class="figcenter"> + <p id="FIG_12" class="f110"><span class="smcap">Fig. 12.</span></p> + <img src="images/i_184.jpg" alt="" width="400" height="409" > +</div> + +<p>You see, thus, by what various paths we are constantly 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 discoveries, by which the +conclusions are confirmed until we have confidence in them. In this +case another circumstance has contributed—that is, the observation +that this process of transformation, which the theory in question +<span class="pagenum" id="Page_185">[Pg 185]</span> +presupposes, goes on still, though on a smaller scale, seeing that all +stages of that process can still be found to exist.</p> + +<div class="figcenter"> + <p id="FIG_13" class="f110"><span class="smcap">Fig. 13.</span></p> + <img src="images/i_185.jpg" alt="" width="400" height="396" > +</div> + +<p>For as we have already seen, the larger bodies which are already formed +go on increasing with the 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 firmament of fixed stars, according to Sir J. Herschel’s +<span class="pagenum" id="Page_186">[Pg 186]</span> +newest catalogue, over 5,000 nebulous spots, of which those whose light +is sufficiently strong give for the most part a coloured spectrum of +fine bright lines, as they appear in the spectra of the ignited gases. +The nebulæ are partly rounded structures, which are called <i>planetary +nebulæ</i> (<a href="#FIG_12">fig. 12</a>); sometimes wholly irregular in form, as the large +nebula in Orion, represented in <a href="#FIG_13">fig. 13</a>; they are partly annular, as in +the figures in <a href="#FIG_14">fig. 14</a>, 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.</p> + +<div class="figcenter"> + <p id="FIG_14" class="f110"><span class="smcap">Fig. 14.</span></p> + <img src="images/i_186.jpg" alt="" width="400" height="464" > +</div> + +<p><span class="pagenum" id="Page_187">[Pg 187]</span></p> + +<div class="figcontainer"> + <div class="figsub"> + <p id="FIG_15" class="f120"><span class="smcap">Fig. 15.</span></p> + <img src="images/i_187a.jpg" alt="" width="250" height="258" > + </div> + <div class="figsub"> + <p id="FIG_16" class="f120"><span class="smcap">Fig. 16.</span></p> + <img src="images/i_187b.jpg" alt="" width="250" height="258" > + </div> +</div> + +<p>In many nebulæ small stars can be seen, as in figs. <a href="#FIG_15">15</a> +and <a href="#FIG_16">16</a>, 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 nebulæ are only heaps of very fine stars, of other Milky Ways. +Now, however, spectrum analysis has shown a gas spectrum in many +nebulæ which contains stars, while actual heaps of stars show the +continuous spectrum of ignited solid bodies. Nebulæ have in general +three distinctly recognisable lines, one of which, in the blue, belongs +to hydrogen, a second in bluish-green to nitrogen,⁠<a id="FNanchor_27_27" href="#Footnote_27_27" class="fnanchor">[27]</a> +<span class="pagenum" id="Page_188">[Pg 188]</span> +while the third, between the two, is of unknown origin. <a href="#FIG_17">Fig. 17</a> +shows 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 <a href="#FIG_17">fig. 17</a>, 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 therefore greatly enfeebled or even extinguished, while the +undecomposable light of bright gas lines remains undecomposed, and +hence can still be seen. In any case, the decomposition of the light +of the nebulæ 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 nebulæ 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 +<span class="pagenum" id="Page_189">[Pg 189]</span> +have been known and observed, they must either have very little mass, +or they must be of colossal size and distance. The former does not +appear very probable, because small masses very soon give out their +heat, and hence we are left to the second alternative, that they are of +huge dimensions and distances. The same conclusion had been originally +drawn by Sir W. Herschel, on the assumption that the nebulæ were heaps +of stars.</p> + +<div class="figcenter"> + <p id="FIG_17" class="f110"><span class="smcap">Fig. 17.</span></p> + <img src="images/i_188.jpg" alt="" width="600" height="103" > +</div> + +<p>With those nebulæ which, besides the lines of gases, also show the +continuous spectrum of ignited denser bodies, are connected spots which +are partly irresolvable and partly resolvable into heaps of stars, +which only show the light of the latter kind.</p> + +<p>The countless luminous stars of the heavenly firmament, 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 quantity of individual elements.</p> + +<p>But we find also in space a third stadium, that of extinct suns; +and for this also there are actual evidences. 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 +<span class="pagenum" id="Page_190">[Pg 190]</span> +still like a fixed star, and finally 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.</p> + +<p>In other cases obscure heavenly bodies have discovered themselves by +their attraction on adjacent bright stars, and the motions of the +latter thereby produced. 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.</p> + +<p>Thus there are extinct suns. The fact that there are such lends new +<span class="pagenum" id="Page_191">[Pg 191]</span> +weight to the reasons which permit 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.</p> + +<p>The term of 17,000,000 years which I have given may perhaps become +considerably prolonged by the gradual abatement of radiation, by the +new accretion of falling meteors, and by still greater condensation +than that which I have assumed in that calculation. But we know of no +natural process which could spare our sun the fate which has manifestly +fallen upon other suns. This is a thought which we only reluctantly +admit; it seems to us an insult to the beneficent Creative Power which +we otherwise find at work in organisms and especially in living ones. +But we must reconcile ourselves to the thought that, however we may +consider ourselves to be the centre and final object of Creation, we +are but as dust on the earth; which again is but a speck of dust in +the immensity of space; and the previous duration of our race, even +if we follow it far beyond our written history, into the era of the +lake dwellings or of the mammoth, is but an instant compared with +the primeval times of our planet; when living beings existed upon +it, whose strange and unearthly remains still gaze at us from their +ancient tombs; and far more does the duration of our race sink into +insignificance compared with the enormous periods during which worlds +<span class="pagenum" id="Page_192">[Pg 192]</span> +have been in 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.</p> + +<p>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? Considering 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 +equilibrium, will not consider 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 +<span class="pagenum" id="Page_193">[Pg 193]</span> +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 dwelling-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.</p> + +<p>However this may be, that which most arouses our moral feelings at +the thought of a future, though possibly 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 inheritance 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. But even +<span class="pagenum" id="Page_194">[Pg 194]</span> +men of such free and large order of minds as Lessing and David Strauss +could not reconcile themselves to the thought of a final destruction of +the living race, and with it of all the fruits of all past generations.</p> + +<p>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 continues to exist—indestructible, imperishable, as an +invisible motion, now agitating the molecules of ponderable 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 +spectroscope. United afresh, these rays may ignite a new flame, and +thus, as it were, acquire a new bodily existence. +<span class="pagenum" id="Page_195">[Pg 195]</span></p> + +<p>Just as the flame remains the same in appearance, and continues to +exist with the same form and structure, 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 continuance of the individual is attached. For the material +of the body, like that of the flame, is subject to continuous 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? +<span class="pagenum" id="Page_196">[Pg 196]</span></p> + +<h3>ADDENDUM.</h3> + +<p>The sentences on <a href="#Page_193">page 193</a> 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:—</p> + +<p>‘I will mention here a further 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 +unscientific.” 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 Heidelberg and Cologne, but not published. I cannot +object if anyone 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.</p> + +<p>‘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 +<span class="pagenum" id="Page_197">[Pg 197]</span> +with a whole skin in the smash-up of its mother-body, and had not +shared the general rise of temperature, it must necessarily have first +passed through the atmosphere of the earth, before it could deliver +itself of its organisms for the purpose of peopling the earth.”</p> + +<p>‘Now, in the first place, we know from repeated observations 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 fragments, which at the first moment were scattered +with planetary 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.</p> + +<p>‘As I have already remarked I am not inclined to suggest that all these +possibilities are probabilities. They are questions the existence and +signification of which we must remember, in order that if the case arise +they may be solved by actual observations or by conclusions therefrom.’</p> + +<p><span class="pagenum"><a id="Page_198">[Pg 198]</a></span></p> + +<hr class="chap x-ebookmaker-drop"> +<div class="chapter"> +<p><span class="pagenum"><a id="Page_199">[Pg 199]</a></span></p> + <h2 class="nobreak">ON THOUGHT IN MEDICINE.</h2> +</div> + +<div class="blockquot"> +<p class="center"><i>An Address delivered August 2, 1877, on +the Anniversary of the Foundation of the Institute for the Education of +Army Surgeons.</i></p> +</div> + +<p>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 judged more +<span class="pagenum" id="Page_200">[Pg 200]</span> +favourably of my Essay than I did myself, and I still possess the books +which were awarded to me as the prize.</p> + +<p>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 developed. Much indeed has +been accomplished.</p> + +<p>Although all that we hoped for has not been fulfilled, and many things +have turned out differently from what we wished, yet we have gained +much for which we could not have dared to hope. Just as the history of +the world has made one of its few giant steps before our eyes, so also +has our science; hence an old student, like myself, scarcely recognises +the somewhat matronly aspect of Dame Medicine, when he accidentally +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.</p> + +<p>I may, perhaps, retain the impression of this antagonism, 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 +<span class="pagenum" id="Page_201">[Pg 201]</span> +for speaking to you about the metamorphosis which has taken place in +medicine during this period, and with the results of whose development +you are better acquainted than I am. I should like the impression +of this development and of its causes not to be quite lost on the +younger of my hearers. They have no special incentive for consulting +the literature of that period; they would meet with principles which +appear as if written in a lost tongue, so that it is by no means easy +for us to transfer ourselves into the mode of thought of a period +which is so far behind us. The course of development of medicine is an +instructive lesson on the true principles of scientific inquiry, and +the positive part of this lesson has, perhaps, in no previous time been +so impressively taught as in the last generation.</p> + +<p>The task falls to me, of teaching that branch of the natural sciences +which has to make the widest generalisations, and has to discuss the +meaning of fundamental 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.</p> + +<p>As regards my acquaintance with the tone of thought of the older +medicine, independently of the general obligation, incumbent on every +<span class="pagenum" id="Page_202">[Pg 202]</span> +educated physician, of understanding the literature of his science and +the direction as well as the conditions of its progress, there was in +my case a special incentive. In my first professorship at Königsberg, +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.</p> + +<p>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.</p> + +<p>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 +therapeutics, arranged it very thoroughly and methodically and with +great critical acumen.</p> + +<p>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 +<span class="pagenum" id="Page_203">[Pg 203]</span> +in which any one who was even moderately at home in physical +considerations found a fruitful virgin 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.</p> + +<p>Perhaps only he can appreciate the immense importance 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?</p> + +<p>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 +commotion. Or he may consider ancient prejudices to be interesting and +pardonable, as remains of poetic romance, 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 +<span class="pagenum" id="Page_204">[Pg 204]</span> +severe tests; he can only use the hard and clear light of facts, and +must give up the notion of lulling himself in agreeable illusions.</p> + +<p>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 understood by +his native land.</p> + +<p>If I am called upon to designate in one word the fundamental error +of that former time, I should be inclined 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 practically an exceedingly important +object of true philosophy.</p> + +<p>The proud word of Hippokrates,</p> + +<p class="f120"><b>ἰητρὸς φιλόσοφος ἰσόθεος,</b></p> + +<p class="center no-wrap">‘Godlike is the physician who is a philosopher’, served,<br> + as it were, as a banner of the old deductive medicine.</p> + +<p><span class="pagenum" id="Page_205">[Pg 205]</span> +We may admit this if only we once agree what we 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 +philosophical or metaphysical considerations. If, therefore, we are +to understand the medical philosopher of Hippokrates to be a man who +has a <i>perfected</i> 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.</p> + +<p>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?</p> + +<p>But those disciples of medicine who thought themselves 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 φιλόσοφος 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 completeness +<span class="pagenum" id="Page_206">[Pg 206]</span> +of the system. It is not difficult to understand how in periods of +youthful development, such a one-sided over-estimate of thought could +be arrived at. The superiority of man over animals, of the scholar over +the barbarian, depends upon thinking; sensation, feeling, perception, +on the contrary, he shares with his lower fellow-creatures, and in +acuteness of the senses many of these are even superior to him. That +man strives to develop his thinking faculty to the utmost is a problem +on the solution of which the feeling of his own dignity, as well as of +his own practical power, depends; and it is a natural error to have +considered unimportant the dowry of mental capacities which Nature had +given to animals, and to have believed that thought could be liberated +from its natural basis, observation and perception, to begin its +Icarian flight of metaphysical speculation.</p> + +<p>It is, in fact, no easy problem to ascertain completely 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 +<span class="pagenum" id="Page_207">[Pg 207]</span> +if the tradition has 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 contented 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.</p> + +<p>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 accustomed very often to mention together the corresponding +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 number 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, we assign +<span class="pagenum" id="Page_208">[Pg 208]</span> +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.</p> + +<p>The adult, moreover, when he begins to reflect upon the origin of his +knowledge, is in possession of a huge mass of everyday 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 conformity +with law is always repeated with regularity, these deeply impressed +remains of all previous conceptions are just the conceptions of what is +conformable to law in the things and processes.</p> + +<p>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 +<span class="pagenum" id="Page_209">[Pg 209]</span> +has possessed as long as he can remember. We need not refer even to the +possibility of inheritance by procreation.</p> + +<p>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 +themselves, the world of facts seems to him, like his conceptions, to +be governed by intellectual forces. We recognise this psychological +anthropomorphism, from the <i>Ideas</i> of Plato, to the immanent +dialectic of the cosmical process of Hegel, and to the unconscious will +of Schopenhauer.</p> + +<p>Natural science, which in former times was virtually identical with +medicine, followed the path of philosophy; 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.</p> + +<p>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, independently of their +<span class="pagenum" id="Page_210">[Pg 210]</span> +theories. Then came the less gifted pupils, who copied their master, +exaggerated his theory, made it more one-sided and more logical, +without regard to any discordance with Nature. The more rigid the +system, the fewer and the more thorough were the methods to which the +healing art was restricted. The more the schools were driven into a +corner by the increase in actual knowledge, the more did they depend +upon the ancient authorities, and the more intolerant were they +against innovation. The great reformer of anatomy, Vesalius, was cited +before the Theological faculty of Salamanca; Servetus was burned at +Geneva along with his book, in which he described the circulation of +the lungs; and the Paris faculty prohibited the teaching of Harvey’s +doctrine of the circulation of the blood in its lecture rooms.</p> + +<p>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 +<i>strictum</i> and <i>laxum</i>, from tone and want of tone, and +afterwards from strained or relaxed nerves and from obstructions in the +<span class="pagenum" id="Page_211">[Pg 211]</span> +vessels. Humoral pathology was only acquainted with alterations in +mixture. The four cardinal fluids, representatives of the classical +four elements, blood, phlegm, black and yellow gall; with others, the +acrimonies or dyscrasies, which had to be expelled by sweating and +purging; in the beginning of our modern epoch, the acids and alkalies +or the alchymistic spirits, and the occult qualities of the substances +assimilated—all these were the elements of this chemistry. Along with +these were found all kinds of physiological conceptions, some of which +contained remarkable foreshadowings, such as the <b>ἔμφυτον θέρμον</b>, +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 <b>πνεῦμα</b>, 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. +<span class="pagenum" id="Page_212">[Pg 212]</span> +Any careful observation 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.</p> + +<p>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 disease, and their +cure, upon any one such simple explanation. 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 +<span class="pagenum" id="Page_213">[Pg 213]</span> +certainty than the principle from which it is deduced; and that each +new induction must in the first place be a new test, by experience, of +its own bases. That a conclusion is deduced by the strictest logical +method from an uncertain premise does not give it a hair’s breadth of +certainty or of value.</p> + +<p>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 <i>wished</i> 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 intolerance partly +<span class="pagenum" id="Page_214">[Pg 214]</span> +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, Aretæus, Galen, Sydenham, and +Boerhaave, had become eclectics, or at any rate very lax systematists.</p> + +<p>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 <i>stimulus</i>; he called the altered condition of +the nerve the <i>excitation</i>, and its capacity of responding to a +stimulus the <i>excitability</i>, which was lost at death. This entire +<span class="pagenum" id="Page_215">[Pg 215]</span> +condition of things, which physically speaking asserts 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 nourishment were considered to be the +normal stimuli. The <i>kind</i> of activity seemed, on the contrary, to +be conditioned 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 treatment 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.</p> + +<p>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’, <i>anima inscia</i>, +<span class="pagenum" id="Page_216">[Pg 216]</span> +in Georg Ernst Stahl, who, in the first half of the last +century, was professor of chemistry and pathology in Halle. Stahl had +a clear and acute mind, which is informing and stimulating, from the +way in which he states the proper question, even in those cases in +which he decides against our present views. He it is who established +the first comprehensive system of chemistry, that of phlogiston. If +we translate his phlogiston into latent heat, the theoretical bases +of his system passed essentially into the system of Lavoisier; Stahl +did not then know oxygen, which occasioned some false hypotheses; for +instance, on the negative gravity of phlogiston. Stahl’s ‘soul of life’ +is, on the whole, constructed on the pattern on which the pietistic +communities of that period represented to themselves the sinful human +soul; it is subject to errors and passions, to sloth, fear, impatience, +sorrow, indiscretion, despair. The physician must first appease it, or +then incite it, or punish it, and compel it to repent. And the way in +which, at the same time, he established the necessity of the physical +and vital actions was well thought out. The soul of life governs the +body, and only acts by means of the physico-chemical forces of the +substances assimilated. But it has the power to bind and to loose +these forces, to allow them full play or to restrain them. After +death the restrained forces become free, and evoke putrefaction and +<span class="pagenum" id="Page_217">[Pg 217]</span> +decomposition. For the refutation of this hypothesis of binding and +loosing, it was necessary to discover the law of the conservation of +force.</p> + +<p>The second half of the previous century was too much possessed by +the principles of rationalism to recognise openly Stahl’s ‘soul of +life.’ It was presented more scientifically as vital force, <i>Vis +vitalis</i>, 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.</p> + +<p>The doctrine of vital force entered into the pathological 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 <i>symptomata morbi</i>, from +those which brought on the reaction of vital force, the <i>symptomata +reactionis</i>. The latter were principally 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 according to +circumstances.</p> + +<p>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 treatment fell comparatively into the background. The +therapeutics of febrile diseases had thereby become very monotonous, +although the means indicated by theory were still abundantly used, and +<span class="pagenum" id="Page_218">[Pg 218]</span> +especially blood-letting, which since that time has almost been +entirely abandoned. 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.</p> + +<p>What we learned at that time were only the ruins of the older +dogmatism, but their doubtful features soon manifested themselves.</p> + +<p>The vitalistic physician considered that the essential 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?</p> + +<p>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 maintained that they were a coarse mechanical means of +<span class="pagenum" id="Page_219">[Pg 219]</span> +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 treating 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, however, were good, and he did not need it.</p> + +<p>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 experiment. +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 +professor of therapeutics, who occupied himself much with the +reorganisation of the Universities, was urgent with me to divide +<span class="pagenum" id="Page_220">[Pg 220]</span> +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.</p> + +<p>I mention these points, which I myself have experienced, 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.</p> + +<p>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 foundation of the chemical laboratory, +was complete, as far as chemistry was concerned, but his example had +<span class="pagenum" id="Page_221">[Pg 221]</span> +not been imitated elsewhere. Yet medicine possessed in anatomical +dissections a great means of education for independent observation, +which is wanting in the other faculties, and to which I am disposed +to attach great weight. Microscopic demonstrations were isolated +and infrequent in the lectures. Microscopic instruments were costly +and scarce. I came into possession of one by having spent my autumn +vacation in 1841 in the Charité, prostrated by typhoid fever; as pupil, +I was nursed without expense, and on my recovery 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 prolongations +of the ganglionic cells in the invertebrata, which I described in +my dissertation, and to investigate the vibrions in my research on +putrefaction and fermentation.</p> + +<p>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 exhaust all possibilities, in the +consideration, until a practicable 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. +<span class="pagenum" id="Page_222">[Pg 222]</span></p> + +<p>It was one man more especially who aroused our enthusiasm for work in +the right direction—that is, Johannes Müller, 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 hypotheses, 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.</p> + +<p>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 +<span class="pagenum" id="Page_223">[Pg 223]</span> +in regard to the sensible energies he not only established the general +law, but carried out a great number of separate investigations, to +eliminate objections, and to refute false indications and evasions. +That which hitherto had been imagined from the data of everyday +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 Müller’s hands in a state of classical perfection—a +scientific achievement whose value I am inclined to consider as equal +to that of the discovery of the law of gravitation.</p> + +<p>His scientific tendency, and more especially his example, 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, Brücke, Ludwig, Traube, J. Meyer, Lieberkühn, Hallmann; we +were succeeded by A. von Graefe, W. Busch, Max Schultze, A. Schneider.</p> + +<p>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 Müller’s +<span class="pagenum" id="Page_224">[Pg 224]</span> +contemporaries, among whom the three brothers Weber of Leipzig must +first of all be mentioned, who have built solid foundations in the +mechanism of the circulation, of the muscles, of the joints, and of the +ear.</p> + +<p>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 particularly by the use of anæsthetic ether +and of the paralysing 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 possibilities of delicate and yet certain diagnosis +where there seemed to be absolute darkness. The continually increasing +number of proved parasitical organisms substitute tangible objects for +mystical entities, and teach the surgeon to forestall the fearfully +subtle diseases of decomposition.</p> + +<p>But do not think, gentlemen, that the struggle is at an end. As long as +there are people of such astounding conceit as to imagine that they can +effect, by a few clever strokes, that which man can otherwise only hope +<span class="pagenum" id="Page_225">[Pg 225]</span> +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.</p> + +<p>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.</p> + +<p>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 +<span class="pagenum" id="Page_226">[Pg 226]</span> +that course. In order to acquire this foreknowledge of what is coming, +but of what has not been settled by observations, no other method is +possible than that of endeavouring to arrive at the laws of facts by +observations; 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 philosopher +reflects at each new phenomenon, whether the best established laws of +the best known forces may not experience a change; it can of course +only be a question of a change which does not contradict the whole +store of our previously collected experiences. It never thus attains +unconditional truth, but such a high degree of probability that it is +practically equal to certainty. The metaphysicians may amuse themselves +at this; we will take their mocking to heart when they are in a +position to do better, or even as well. The old words of Socrates, the +prime master of inductive definitions, in reference to them are just as +fresh as they were 2,000 years ago: ‘They imagined they knew what they +<span class="pagenum" id="Page_227">[Pg 227]</span> +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 (<b>τοῖς μαινομένοις ὁμοίως</b>).⁠<a id="FNanchor_28_28" href="#Footnote_28_28" class="fnanchor">[28]</a> +Socrates calls them <b>τοὺς μέγιστον φρονοῦντας</b>. Schopenhauer⁠<a id="FNanchor_29_29" href="#Footnote_29_29" class="fnanchor">[29]</a> +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.</p> + +<p>In speaking against the empty manufacture of hypotheses, 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 +<span class="pagenum" id="Page_228">[Pg 228]</span> +cannot be acquired by any known method. 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 contemporaries. 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 developed—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 +perceptions.</p> + +<p>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 <i>always</i> 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 +<span class="pagenum" id="Page_229">[Pg 229]</span> +workers who are shy at bringing 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.</p> + +<p>In the ‘type case’ of the printer all the wisdom of the world is +contained which has been or can be discovered; 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 hypotheses are exhausted, and among these there must +necessarily be many fragments of the correct theory. But who knows how +to find them?</p> + +<p>I insist upon this in order to make clear to you that all this +literature, of untried and unconfirmed hypotheses, has no value in the +progress of science. On the contrary, the few sound ideas which they +may contain are concealed by the rubbish of the rest; and one who wants +to publish something really new—facts—sees himself open to the danger +of countless claims of priority, unless he is prepared to waste time +and power in reading beforehand a quantity of absolutely useless books, +<span class="pagenum" id="Page_230">[Pg 230]</span> +and to destroy his readers’ patience by a multitude of useless +quotations.</p> + +<p>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 <i>Kritik der reinen Vernunft</i> 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 metaphysics was striving after; and hence geometrical +axioms, which he looked upon as <i>à priori</i> 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 <i>à priori</i> intuition has been the anchoring-ground of +metaphysicians. 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 perception of the senses is the expression of this theory in +physiology. All mathematicians united to fight against any attempt to +<span class="pagenum" id="Page_231">[Pg 231]</span> +resolve the intuitions into their natural elements; whether the +so-called pure or the empirical, the axioms of geometry, the principles +of mechanics, or the perceptions of vision. For this reason, therefore, +the mathematical investigations of Lobatschewsky, Gauss, and Riemann on +the alterations 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 experience—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 +investigators who endeavour to eliminate from among the perceptions of +the senses, whatever there may be of the actions of memory, and of the +repetition of similar impressions, 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 prolongation of ganglionic +cells, and of the connection of nerve-fibres in the brain. +<span class="pagenum" id="Page_232">[Pg 232]</span></p> + +<p>Indeed, however self-evident, and however important the principle may +appear to be, that natural science has to seek for the laws of facts, +this principle is nevertheless often forgotten. In recognising the law +found, as a force which rules the processes in nature, we conceive it +objectively as a <i>force</i>, and such a reference of individual cases +to a force which under given conditions produces a definite result, +that we designate as a causal explanation of phenomena. We cannot +always refer to the forces of atoms; we speak of a refractive force, +of electromotive and of electrodynamic force. But do not forget the +<i>given conditions</i> and the <i>given result</i>. 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.</p> + +<p>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 Müller gave +to the idea of reflex action, is gradually evaporated into this, that +when an impression 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 +<span class="pagenum" id="Page_233">[Pg 233]</span> +upon the irresolvable complexity of the nerve-fibres of the brain. But +the resemblance to the <i>qualitates occultæ</i> of ancient medicine is +very suspicious.</p> + +<p>From the entire chain of my argument it follows that what I have +said against metaphysics is not intended against philosophy. But +metaphysicians have always tried to plume themselves on being +philosophers, and philosophical amateurs have mostly taken an interest +in the high-flying speculations of the metaphysicians, 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<a id="FNanchor_30_30" href="#Footnote_30_30" class="fnanchor">[30]</a> +I compared the relationship 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 connoisseurs 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.</p> + +<p>In like manner, philosophy, if it gives up metaphysics, 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 insight into the +<span class="pagenum" id="Page_234">[Pg 234]</span> +action of his optical instrument, in like manner every scientific +enquirer must study minutely the chief instrument of his research as +to its capabilities. The groping of the medical schools for the last +two thousand years is, among other things, an illustration 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 activity. 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.</p> + +<p>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 opponents, 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. +<span class="pagenum" id="Page_235">[Pg 235]</span></p> + +<p>My own researches have led me more than other disciples of the school +of natural science into controversial regions; and the expressions of +metaphysical discontent have perhaps concerned me even more than my +friends, as many of you are doubtless aware.</p> + +<p>In order, therefore, to leave my own personal opinions quite on +one side, I have allowed two unsuspected warrantors to speak for +me—Socrates and Kant—both of whom were certain that all metaphysical +systems established up to their time were full of empty false +conclusions, 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.</p> + +<p>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 +<span class="pagenum" id="Page_236">[Pg 236]</span> +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 +knowledge.’</p> + +<p>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 nonsense 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 knowledge of nature, you are those who begin +with the best mental preparation, and are acquainted with the most +varied regions of natural phenomena.</p> + +<p>In order, finally, to conclude our consultation on the condition of +Dame Medicine correctly with the epikrisis, 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.</p> + +<hr class="chap x-ebookmaker-drop"> +<div class="chapter"> +<p><span class="pagenum" id="Page_237">[Pg 237]</span></p> + <h2 class="nobreak">ON ACADEMIC FREEDOM IN<br> GERMAN UNIVERSITIES.</h2> +</div> + +<p class="center no-wrap"><i>Inaugural Address as Rector of the<br> Frederick William University of +Berlin.<br> Delivered October 15, 1877.</i></p> + +<p>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 necessitated many changes in the old order +of University teaching, and which will, perhaps, necessitate other +changes. It is indeed just in that branch (Physics) which I represent, +<span class="pagenum" id="Page_238">[Pg 238]</span> +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 several times in the +position of having to propose alterations 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 +thoughtless 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 Rector at Heidelberg, the +closest connection 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 +<span class="pagenum" id="Page_239">[Pg 239]</span> +natural sciences far more prominently—that the real is far more +sharply separated from the unreal, by the incorruptible criticism +of facts, than is the case with the more complex problems of mental +science.</p> + +<p>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 required. We can scarcely foresee +what fresh demands and what new problems we may have to meet in the +more immediate future.</p> + +<p>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.</p> + +<p>Under these circumstances it is our duty to get a clear understanding +of the reason for the previous prosperity of our Universities; we must +<span class="pagenum" id="Page_240">[Pg 240]</span> +try to find what is the feature in their arrangements which we must +seek to retain as a precious jewel, and where, on the contrary, we +may give way when changes are required. I consider myself by no means +entitled to give a final opinion on this matter. The point of view of +any single individual 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.</p> + +<p>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 +degrees. The students of that time were mostly men of mature years, who +frequented the University more immediately for their own instruction +and without any direct practical object; but younger men soon began to +be sent, who, for the most part, were placed under the superintendence +of the older members. The separate Universities split again into closer +economic unions, under the name of ‘Nations,’ ‘Bursaries,’ ‘Colleges,’ +<span class="pagenum" id="Page_241">[Pg 241]</span> +whose older members, the seniors, governed the common affairs of each +such union, and also met together for regulating the common affairs +of the University. In the courtyard of the University of Bologna are +still to be seen the coats-of-arms, and lists of members and seniors, +of many such Nations in ancient times. The older graduated members were +regarded as permanent life members of such Unions, and they retained +the right of voting, as is still the case in the College of Doctors +in the University of Vienna, and in the Colleges of Oxford and of +Cambridge, or was until recently.</p> + +<p>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 compelled to quit +the University in a body. This was not restricted to those cases in +which the Church intermeddled, and where political or metaphysical +propositions were in question. Even the medical faculties—that of +<span class="pagenum" id="Page_242">[Pg 242]</span> +Paris, the most celebrated of all at the head—allowed no divergence +from that which they regarded as the teaching of Hippocrates. Anyone +who used the medicines of the Arabians or who believed in the +circulation of the blood was expelled.</p> + +<p>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.</p> + +<p>Until lately, it might have been said that the least change has +taken place in the old English Universities, Oxford and Cambridge. +Their great endowments, the political feeling of the English for the +retention 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 Roman and now of the Anglican +Church, whose instruction laymen might also share in so far as it could +serve the general education of the mind; they were subjected to such +a control and mode of life, as was formerly considered to be good for +young priests. They lived, as they still live, in colleges, under the +<span class="pagenum" id="Page_243">[Pg 243]</span> +superintendence of a number of older graduate members (Fellows) of the +College; in other respects in the style and habits of the well-to-do +classes in England.</p> + +<p>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 Repetitoria which are here +and there held in our Universities. 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 academical degrees are acquired.</p> + +<p>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 <i>may</i>, but <i>need not</i> act +<span class="pagenum" id="Page_244">[Pg 244]</span> +as tutors for the 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.</p> + +<p>In certain respects the English Universities do a great deal. They +bring up their students as cultivated 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 +<span class="pagenum" id="Page_245">[Pg 245]</span> +writing which shows itself in the way in which they handle their +mother tongue. I fear that one of the weakest sides in the instruction +of German youth is in this direction. In the second place the English +Universities, like their schools, take greater care of the bodily +health of their students. They live and work in airy, spacious +buildings, surrounded 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 <i>moral</i> +effect of the more rigorous control is said to be rather illusory.</p> + +<p>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.</p> + +<p>The development of French Universities has been quite different, and +<span class="pagenum" id="Page_246">[Pg 246]</span> +indeed almost in the opposite direction. In accordance with the +tendency of the French to throw overboard everything of historic +development 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 distinct from those institutions which are to further +the progress of science, such as the <i>Collège de France</i>, the +<i>Jardin des Plantes</i>, and the <i>École des Études Supérieures</i>. +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 excite 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 +instruction 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 <i>in verba +magistri</i>; this gives a happy self-satisfaction and freedom from +doubts. If the teacher has been well chosen, this is sufficient in +<span class="pagenum" id="Page_247">[Pg 247]</span> +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 defects in their +system of teaching.</p> + +<p>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 contribution 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.</p> + +<p>Outside the lecture-rooms, the French students live without control, +and associate with young men of other callings, without any special +<i>esprit de corps</i> or common feeling.</p> + +<p>The development of the German Universities differs characteristically +from these two extremes. Too poor in their own possessions not to be +<span class="pagenum" id="Page_248">[Pg 248]</span> +compelled, with increasing demands for the means of instruction, +eagerly to accept the help of the State, and too weak to resist +encroachments upon their ancient rights in times in which modern +States attempt to consolidate themselves, 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 remained attached to it. It is +very remarkable how among wars and political changes in the States +fighting with the decaying Empire for the consolidation of their young +sovereignties, while almost all other privileged 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. +<span class="pagenum" id="Page_249">[Pg 249]</span></p> + +<p>We have retained the old conception of students, as that of young men +responsible to themselves, striving after science of their own free +will, and to whom it is left to arrange their own plan of studies as +they think best. If attendance on particular lectures was enjoined +for certain callings—what are called ‘compulsory lectures’—these +regulations were not made by the University, 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 essential part of study.</p> + +<p>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, +<span class="pagenum" id="Page_250">[Pg 250]</span> +with their own jurisdiction, which extended to the right over life +and death of their own members. As they lived for the most part on +foreign soil, it was necessary to have their own jurisdiction, partly +to protect the members from the caprices of foreign judges, partly to +keep up that degree of respect and order, within the society, which +was necessary to secure the continuation of the rights of hospitality +on a foreign soil; and partly, again, to settle disputes among the +members. In modern times the remains of this academic jurisdiction +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 guarantee 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 +considered fortunate that German students have retained a vivid sense +of corporate union, and of what is intimately connected therewith, +a requirement of honourable 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. +<span class="pagenum" id="Page_251">[Pg 251]</span></p> + +<p>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 expressions +of this feeling. Nothing of this kind is but even faintly suggested +in the literature of other European 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.</p> + +<p>When I think of my own student life, and of the impression which a man +like Johannes Müller, 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 +<span class="pagenum" id="Page_252">[Pg 252]</span> +mental standard altered for the rest of his life. Such intercourse is, +moreover, the most interesting that life can offer.</p> + +<p>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 freedom, 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.</p> + +<p>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 instruction given, if possible, +<span class="pagenum" id="Page_253">[Pg 253]</span> +only by teachers who have proved their own power of advancing science. +This also is a point in respect to which the English and French often +express their surprise. They lay more weight than the Germans on what +is called the ‘talent for teaching’—that is, the power of explaining +the subjects of instruction in a well-arranged and clear manner, and, +if possible, with eloquence, and so as to entertain and to fix the +attention. Lectures of eloquent orators at the Collège de France, +Jardin des Plantes, as well as in Oxford and Cambridge, are often the +centres of the elegant and the educated world. In Germany we are not +only indifferent to, but even distrustful of, oratorical ornament, +and often enough are more negligent than we should be of the outer +forms of the lecture. There can be no doubt that a good lecture can +be followed with far less exertion than a bad one; that the matter +of the first can be more certainly and completely apprehended; that +a well-arranged explanation, which develops the salient points and +the divisions of the subject, 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, +<span class="pagenum" id="Page_254">[Pg 254]</span> +and hesitating delivery. Yet I have not infrequently seen that such +teachers had crowded lecture-rooms, while empty-headed orators excited +astonishment in the first lecture, fatigue in the second, and were +deserted in the third. Anyone who desires to give his hearers a perfect +conviction of the truth of his principles must, first of all, know +from his own experience how conviction is acquired and how not. He +must have known how to acquire conviction 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.</p> + +<p>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 +<span class="pagenum" id="Page_255">[Pg 255]</span> +of knowledge in so far as these consist in books and monuments, or in +experiments, and in the observation of natural objects and processes.</p> + +<p>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 +beginning 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.</p> + +<p>The free conviction of the student can only be acquired when freedom +of expression is guaranteed to the teacher’s own conviction—the +<i>liberty of teaching</i>. 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 consequences of materialistic metaphysics, the +boldest speculations upon the basis of Darwin’s theory of evolution, +may be taught in German Universities with as little restraint as the +most extreme deification of Papal Infallibility. As in the tribune of +<span class="pagenum" id="Page_256">[Pg 256]</span> +European Parliaments 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 forbidden. 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 Collège de France the +lectures of a man of Renan’s scientific importance and earnestness are +forbidden.</p> + +<p>I have to speak of another aspect of our liberty of teaching. That is, +the extended sense in which German Universities have admitted teachers. +In the original meaning of the word, a doctor is a ‘teacher,’ or one +whose capacity as teacher is recognised. In the Universities of the +Middle Ages any doctor who found pupils could set up as teacher. In +course of time the practical signification of the title was changed. +Most of those who sought the title did not intend to act as teachers, +but only needed it as an official recognition of their scientific +training. Only in Germany are there any remains of this ancient right. +In accordance with the altered meaning of the title of doctor, and +the minuter specialisation of the subjects of instruction, a special +proof of more profound scientific proficiency, in the particular branch +in which they wish to habilitate, is required from those doctors who +desire to exercise the right of teaching. In most German Universities, +<span class="pagenum" id="Page_257">[Pg 257]</span> +moreover, the legal status of these habilitated 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 +examination. 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.</p> + +<p>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 uncertain prospects for the future, +devote themselves to strenuous scientific work. And, judging us from +the point of view of basely practical interests, they are equally +<span class="pagenum" id="Page_258">[Pg 258]</span> +surprised that the faculties so readily admit young men who at any +moment may change from assistants to competitors; and further, that +only in the most exceptional cases is anything ever heard of unworthy +means of competition in what is a matter of some delicacy.</p> + +<p>The appointment to vacant professorships, like the admission of +private docents, rests, though not unconditionally, 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. +<span class="pagenum" id="Page_259">[Pg 259]</span></p> + +<p>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 consciousness of having not too small a number of intelligent +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 successful 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.</p> + +<p>With regard to the spectre of rivalry among University 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 there are two +<span class="pagenum" id="Page_260">[Pg 260]</span> +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 scientific 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 distinguished 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.</p> + +<p>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.</p> + +<p>The predominance of a distinct scientific school in a faculty may +become more injurious than such personal 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.</p> + +<p>We see best how strenuously the Universities under this system have +sought to attract the scientific ability of Germany when we consider +<span class="pagenum" id="Page_261">[Pg 261]</span> +how many pioneers have remained outside the Universities. The answer +to such an inquiry is given in the not infrequent jest or sneer that +all wisdom in Germany is professorial wisdom. If we look at England, +we see men like Humphry Davy, Faraday, Mill, Grote, who have had +no connection with English Universities. If, on the other hand, we +deduct from the list of German men of science those who, like David +Strauss, have been driven away by Government 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 Royal Institution of London, a private +Society, which provides for its members 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 institutions which exert the most powerful attraction +<span class="pagenum" id="Page_262">[Pg 262]</span> +on the taught. But it is clear that this attraction depends on the +teacher’s hope that he will not only find in the University a body of +pupils enthusiastic and accustomed 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.</p> + +<p>The entire organisation of our Universities is thus permeated by +this respect for a free independent conviction, which is more +strongly impressed on the Germans than on their Aryan kindred of the +Celtic and Romanic branches, in whom practical political motives +have greater weight. They are able, and as it would seem with +perfect conscientiousness, to restrain the inquiring mind from the +investigation of those principles which appear to them to be beyond +the range of discussion, as forming the foundation of their political, +social, and religious organisation; they think themselves quite +justified in not allowing their youth to look beyond the boundary which +they themselves are not disposed to overstep.</p> + +<p>If, therefore, any region of questions is to be considered 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 +<span class="pagenum" id="Page_263">[Pg 263]</span> +prescribed path, and teachers must be appointed who do not rebel +against authority. We can then, however, only speak of free conviction +in a very limited sense.</p> + +<p>You see how different was the plan of our forefathers. 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 conviction 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 Rome, also organised the German Universities.</p> + +<p>But any institution based upon freedom must also be able to calculate +on the judgment and reasonableness 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 necessitates 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 +<span class="pagenum" id="Page_264">[Pg 264]</span> +the interest of his hearers. The estimate which he places on his +task will depend on how far he is followed by the appreciation of a +sufficient 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 sufficiently logically trained +judgment, with a sufficient habit of mental exertion, with a tact +sufficiently developed 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 +themselves 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.</p> + +<p>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 frequented them, who were less developed in the above +<span class="pagenum" id="Page_265">[Pg 265]</span> +respects. The general self-respect of the students 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 scrupulous 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 qualifications of an independent +academical teacher.</p> + +<p>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 independence of conviction. I say work; for independence +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 revolted 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.</p> + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_266">[Pg 266]</span></p> +<h2 class="nobreak">HERMANN VON HELMHOLTZ </h2> +</div> + +<p class="f120">AN AUTOBIOGRAPHICAL SKETCH.</p> +<p class="center"><i>An Address delivered on the occasion of his Jubilee, 1891.</i></p> + +<p>In the course of the past year, and most recently on the occasion +of the celebration of my seventieth birthday, and the subsequent +festivities, I have been overloaded with honours, with marks of respect +and of goodwill in a way which could never have been expected. My own +sovereign, his Majesty the German Emperor, has raised me to the highest +rank in the Civil Service; the Kings of Sweden and of Italy, my former +sovereign, the Grand Duke of Baden, and the President of the French +Republic, have conferred Grand Crosses on me; many academies, not only +of science, but also of the fine arts, faculties, and learned societies +spread over the whole world, from Tomsk to Melbourne, have sent me +diplomas, and richly illuminated addresses, expressing in elevated +language their recognition of my scientific endeavours, and their +thanks for those endeavours, in terms which I cannot read without a +<span class="pagenum" id="Page_267">[Pg 267]</span> +feeling of shame. My native town, Potsdam, has conferred its freedom +on me. To all this must be added countless individuals, scientific and +personal friends, pupils, and others personally unknown to me, who have +sent their congratulations in telegrams and in letters.</p> + +<p>But this is not all. You desire to make my name the banner, as it +were, of a magnificent institution which, founded by lovers of science +of all nations, is to encourage and promote scientific inquiry in +all countries. Science and art are, indeed, at the present time +the only remaining bond of peace between civilised nations. Their +ever-increasing development is a common aim of all; is effected by +the common work of all, and for the common good of all. A great and +a sacred work! The founders even wish to devote their gift to the +promotion of those branches of science which all my life I have +pursued, and thus bring me, with my shortcomings, before future +generations almost as an exemplar of scientific investigation. This +is the proudest honour which you could confer upon me, in so much as +you thereby show that I possess your unqualified favourable opinion. +But it would border on presumption were I to accept it without a quiet +expectation on my part that the judges of future centuries will not be +influenced by considerations of personal favour. +<span class="pagenum" id="Page_268">[Pg 268]</span></p> + +<p>My personal appearance even, you have had represented in marble by a +master of the first rank, so that I shall appear to the present and +to future generations in a more ideal form; and another master of the +etching needle has ensured that faithful portraits of me shall be +distributed among my contemporaries.</p> + +<p>I cannot fail to remember that all you have done is an expression of +the sincerest and warmest goodwill on your part, and that I am most +deeply indebted to you for it.</p> + +<p>I must, however, be excused if the first effect of these abundant +honours is rather surprising and confusing to me than intelligible. +My own consciousness does not justify me in putting a measure of the +value of what I have tried to do, which would leave such a balance +in my favour as you have drawn. I know how simply everything I have +done has been brought about; how scientific methods worked out by my +predecessors have naturally led to certain results, and how frequently +a fortunate circumstance or a lucky accident has helped me. But the +chief difference is this—that which I have seen slowly growing from +small beginnings through months and years of toilsome and tentative +work, all that suddenly starts before you like Pallas fully equipped +from the head of Jupiter. A feeling of surprise has entered into your +estimate, but not into mine. At times, and perhaps even frequently, my +<span class="pagenum" id="Page_269">[Pg 269]</span> +own estimate may possibly have been unduly lowered by the fatigue of +the work, and by vexation about all kinds of futile steps which I +had taken. My colleagues, as well as the public at large, estimate a +scientific or artistic work according to the utility, the instruction, +or the pleasure which it has afforded. An author is usually disposed +to base his estimate on the labour it has cost him, and it is but +seldom that both kinds of judgment agree. It can, on the other hand, be +seen from incidental expressions of some of the most celebrated men, +especially of artists, that they lay but small weight on productions +which seem to us inimitable, compared with others which have been +difficult, and yet which appear to readers and observers as much less +successful. I need only mention Goethe, who once stated to Eckermann +that he did not estimate his poetical works so highly as what he had +done in the theory of colours.</p> + +<p>The same may have happened to me, though in a more modest degree, if I +may accept your assurances and those of the authors of the addresses +which have reached me. Permit me, therefore, to give you a short +account of the manner in which I have been led to the special direction +of my work.</p> + +<p>In my first seven years I was a delicate boy, for long confined to +my room, and often even to bed; but, nevertheless, I had a strong +inclination towards occupation and mental activity. My parents busied +themselves a good deal with me; picture books and games, especially with +<span class="pagenum" id="Page_270">[Pg 270]</span> +wooden blocks, filled up the rest of the time. Reading came pretty +early, which, of course, greatly increased the range of my occupations. +But a defect of my mental organisation showed itself almost as +early, in that I had a bad memory for disconnected things. The +first indication of this I consider to be the difficulty I had in +distinguishing between left and right; afterwards, when at school +I began with languages, I had greater difficulties than others in +learning words, irregular grammatical forms, and peculiar terms of +expression. History as then taught to us I could scarcely master. To +learn prose by heart was martyrdom. This defect has, of course, only +increased, and is a vexation of my mature age.</p> + +<p>But when I possessed small mnemotechnical methods, or merely such as +are afforded by the metre and rhyme of poetry, learning by heart, +and the retention of what I had learnt, went on better. I easily +remembered poems by great authors, but by no means so easily the +somewhat artificial verses of authors of the second rank. I think that +is probably due to the natural flow of thought in good poems, and I am +inclined to think that in this connection is to be found an essential +basis of æsthetic beauty. In the higher classes of the Gymnasium I +could repeat some books of the Odyssey, a considerable number of the +odes of Horace, and large stores of German poetry. In other directions +<span class="pagenum" id="Page_271">[Pg 271]</span> +I was just in the position of our older ancestors, who were not able to +write, and hence expressed their laws and their history in verse, so as +to learn them by heart.</p> + +<p>That which a man does easily he usually does willingly; hence I was +first of all a great admirer and lover of poetry. This inclination was +encouraged by my father, who, while he had a strict sense of duty, +was also of an enthusiastic disposition, impassioned for poetry, and +particularly for the classic period of German Literature. He taught +German in the upper classes of the Gymnasium, and read Homer with us. +Under his guidance we did, alternately, themes in German prose and +metrical exercises—poems as we called them. But even if most of us +remained indifferent poets, we learned better in this way, than in any +other I know of, how to express what we had to say in the most varied +manner.</p> + +<p>But the most perfect mnemotechnical help is a knowledge of the laws +of phenomena. This I first got to know in geometry. From the time +of my childish playing with wooden blocks, the relations of special +proportions to each other were well known to me from actual perception. +What sort of figures were produced when bodies of regular shape were +laid against each other I knew well without much consideration. When I +began the scientific study of geometry, all the facts which I had to +<span class="pagenum" id="Page_272">[Pg 272]</span> +learn were perfectly well known and familiar to me, much to the +astonishment of my teachers. So far as I recollect, that came out +incidentally in the elementary school attached to the Potsdam Training +College, which I attended up to my eighth year. Strict scientific +methods, on the contrary, were new to me, and with their help I saw the +difficulties disappear which had hindered me in other regions.</p> + +<p>One thing was wanting in geometry; it dealt exclusively with abstract +forms of space, and I delighted in complete reality. As I became bigger +and stronger I went about with my father and my schoolfellows a great +deal in the neighbourhood of my native town, Potsdam, and I acquired +a great love of Nature. This is perhaps the reason why the first +fragments of physics which I learned in the Gymnasium engrossed me much +more closely than purely geometrical and algebraical studies. Here +there was a copious and multifarious region, with the mighty fulness +of Nature, to be brought under the dominion of a mentally apprehended +law. And, in fact, that which first fascinated me was the intellectual +mastery over Nature, which at first confronts us as so unfamiliar, +by the logical force of law. But this, of course, soon led to the +recognition that knowledge of natural processes was the magical key +which places ascendency over Nature in the hands of its possessor. In +this order of ideas I felt myself at home. +<span class="pagenum" id="Page_273">[Pg 273]</span></p> + +<p>I plunged then with great zeal and pleasure into the study of all +the books on physics I found in my father’s library. They were very +old-fashioned; phlogiston still held sway, and galvanism had not +grown beyond the voltaic pile. A young friend and myself tried, with +our small means, all sorts of experiments about which we had read. +The action of acids on our mothers’ stores of linen we investigated +thoroughly; we had otherwise but little success. Most successful was, +perhaps, the construction of optical instruments by means of spectacle +glasses, which were to be had in Potsdam, and a small botanical lens +belonging to my father. The limitation of our means had at that time +the value that I was compelled always to vary in all possible ways my +plans for experiments, until I got them in a form in which I could +carry them out. I must confess that many a time when the class was +reading Cicero or Virgil, both of which I found very tedious, I was +calculating under the desk the path of rays in a telescope, and I +discovered, even at that time, some optical theorems, not ordinarily +met with in text-books, but which I afterwards found useful in the +construction of the ophthalmoscope.</p> + +<p>Thus it happened that I entered upon that special line of study to +which I have subsequently adhered, and which, in the conditions I have +<span class="pagenum" id="Page_274">[Pg 274]</span> +mentioned, grew into an absorbing impulse, amounting even to a passion. +This impulse to dominate the actual world by acquiring an understanding +of it, or what, I think, is only another expression for the same thing, +to discover the causal connection of phenomena, has guided me through +my whole life, and the strength of this impulse is possibly the reason +why I found no satisfaction in apparent solutions of problems so long +as I felt there were still obscure points in them.</p> + +<p>And now I was to go to the university. Physics was at that time looked +upon as an art by which a living could not be made. My parents were +compelled to be very economical, and my father explained to me that he +knew of no other way of helping me to the study of Physics, than by +taking up the study of medicine into the bargain. I was by no means +averse from the study of living Nature, and assented to this without +much difficulty. Moreover, the only influential person in our family +had been a medical man, the late Surgeon-General Mursinna; and this +relationship was a recommendation in my favour among other applicants +for admission to our Army Medical School, the Friedrich Wilhelms +Institut, which very materially helped the poorer students in passing +through their medical course.</p> + +<p>In this study I came at once under the influence of a profound +<span class="pagenum" id="Page_275">[Pg 275]</span> +teacher—Johannes Müller; he who at the same time introduced E. Du +Bois-Reymond, E. Brücke, C. Ludwig, and Virchow to the study of +anatomy and physiology. As respects the critical questions about the +nature of life, Müller still struggled between the older—essentially +the metaphysical—view and the naturalistic one, which was then being +developed; but the conviction that nothing could replace the knowledge +of facts forced itself upon him with increasing certainty, and it may +be that his influence over his students was the greater because he +still so struggled.</p> + +<p>Young people are ready at once to attack the deepest problems, and thus +I attacked the perplexing question of the nature of the vital force. +Most physiologists had at that time adopted G. E. Stahl’s way out of +the difficulty, that while it is the physical and chemical forces of +the organs and substances of the living body which act on it, there +is an indwelling vital soul or vital force which could bind and loose +the activity of these forces; that after death the free action of +these forces produces decomposition, while during life their action is +continually being controlled by the soul of life. I had a misgiving +that there was something against nature in this explanation; but it +took me a good deal of trouble to state my misgiving in the form of a +definite question. I found ultimately, in the latter years of my career +<span class="pagenum" id="Page_276">[Pg 276]</span> +as a student, that Stahl’s theory ascribed to every living body the +nature of a <i>perpetuum mobile</i>. I was tolerably well acquainted +with the controversies on this latter subject. In my school days I had +heard it discussed by my father and our mathematical teachers, and +while still a pupil of the Friedrich Wilhelms Institut I had helped in +the library, and in my spare moments had looked through the works of +Daniell, Bernouilli, D’Alembert, and other mathematicians of the last +century. I thus came upon the question, ‘What relations must exist +between the various kinds of natural forces for a perpetual motion to +be possible?’ and the further one, ‘Do those relations actually exist?’ +In my essay, ‘On the Conservation of Force,’ my aim was merely to give +a critical investigation and arrangement of the facts for the benefit +of physiologists.</p> + +<p>I should have been quite prepared if the experts had ultimately said, +‘We know all that. What is this young doctor thinking about, in +considering himself called upon to explain it all to us so fully?’ +But, to my astonishment, the physical authorities with whom I came in +contact took up the matter quite differently. They were inclined to +deny the correctness of the law, and in the eager contest in which +they were engaged against Hegel’s Natural Philosophy were disposed +to declare my essay to be a fantastical speculation. Jacobi, the +<span class="pagenum" id="Page_277">[Pg 277]</span> +mathematician, who recognised the connection of my line of thought +with that of the mathematicians of the last century, was the only +one who took an interest in my attempt, and protected me from being +misconceived. On the other hand, I met with enthusiastic applause and +practical help from my younger friends, and especially from E. Du Bois +Reymond. These, then, soon brought over to my side the members of the +recently formed Physical Society of Berlin. About Joule’s researches on +the same subject I knew at that time but little, and nothing at all of +those of Robert Mayer.</p> + +<p>Connected with this were a few smaller experimental researches on +putrefaction and fermentation, in which I was able to furnish a proof, +in opposition to Liebig’s contention, that both were by no means purely +chemical decompositions, spontaneously occurring, or brought about by +the aid of the atmospheric oxygen; that alcoholic fermentation more +especially was bound up with the presence of yeast spores which are +only formed by reproduction. There was, further, my work on metabolism +in muscular action, which afterwards was connected with that on the +development of heat in muscular action; these being processes which +were to be expected from the law of the conservation of force.</p> + +<p>These researches were sufficient to direct upon me the attention of +<span class="pagenum" id="Page_278">[Pg 278]</span> +Johannes Müller as well as of the Prussian Ministry of Instruction, +and to lead to my being called to Berlin as Brücke’s successor, and +immediately thereupon to the University of Königsberg. The Army medical +authorities, with thank-worthy liberality, very readily agreed to +relieve me from the obligation to further military service, and thus +made it possible for me to take up a scientific position.</p> + +<p>In Königsberg I had to lecture on general pathology and physiology. +A university professor undergoes a very valuable training in being +compelled to lecture every year, on the whole range of his science, in +such a manner that he convinces and satisfies the intelligent among his +hearers—the leading men of the next generation. This necessity yielded +me, first of all, two valuable results.</p> + +<p>For in preparing my course of lectures, I hit directly on the +possibility of the ophthalmoscope, and then on the plan of measuring +the rate of propagation of excitation in the nerves.</p> + +<p>The ophthalmoscope is, perhaps, the most popular of my scientific +performances, but I have already related to the oculists how luck +really played a comparatively more important part than my own merit. I +had to explain to my hearers Brücke’s theory of ocular illumination. In +this, Brücke was actually within a hair’s breadth of the invention of +<span class="pagenum" id="Page_279">[Pg 279]</span> +the ophthalmoscope. He had merely neglected to put the question, To +what optical image do the rays belong, which come from the illuminated +eye? For the purpose he then had in view it was not necessary to +propound this question. If he had put it, he was quite the man to +answer it as quickly as I could, and the plan of the ophthalmoscope +would have been given. I turned the problem about in various ways, to +see how I could best explain it to my hearers, and I thereby hit upon +the question I have mentioned. I knew well, from my medical studies, +the difficulties which oculists had about the conditions then comprised +under the name of Amaurosis, and I at once set about constructing +the instrument by means of spectacle glasses and the glass used for +microscope purposes. The instrument was at first difficult to use, and +without an assured theoretical conviction that it must work, I might, +perhaps, not have persevered. But in about a week I had the great joy +of being the first who saw clearly before him a living human retina.</p> + +<p>The construction of the ophthalmoscope had a very decisive influence on +my position in the eyes of the world. From this time forward I met with +the most willing recognition and readiness to meet my wishes on the +part of the authorities and of my colleagues, so that for the future I +was able to pursue far more freely the secret impulses of my desire for +<span class="pagenum" id="Page_280">[Pg 280]</span> +knowledge. I must, however, say that I ascribed my success in great +measure to the circumstance that, possessing some geometrical capacity, +and equipped with a knowledge of physics, I had, by good fortune, +been thrown among medical men, where I found in physiology a virgin +soil of great fertility; while, on the other hand, I was led by the +consideration of the vital processes to questions and points of view +which are usually foreign to pure mathematicians and physicists. Up to +that time I had only been able to compare my mathematical abilities +with those of my fellow-pupils and of my medical colleagues; that I was +for the most part superior to them in this respect did not, perhaps, +say very much. Moreover, mathematics was always regarded in the school +as a branch of secondary rank. In Latin composition, on the contrary, +which then decided the palm of victory, more than half my fellow-pupils +were ahead of me.</p> + +<p>In my own consciousness, my researches were simple logical applications +of the experimental and mathematical methods developed in science, +which by slight modifications could be easily adapted to the particular +object in view. My colleagues and friends, who, like myself, had +devoted themselves to the physical aspect of physiology, furnished +results no less surprising.</p> + +<p>But in the course of time matters could not remain in that stage. +<span class="pagenum" id="Page_281">[Pg 281]</span> +Problems which might be solved by known methods I had gradually to hand +over, to the pupils in my laboratory, and for my own part turn to +more difficult researches, where success was uncertain, where general +methods left the investigator in the lurch, or where the method itself +had to be worked out.</p> + +<p>In those regions also which come nearer the boundaries of our knowledge +I have succeeded in many things experimental and mechanical—I do not +know if I may add philosophical. In respect of the former, like any one +who has attacked many experimental problems, I had become a person of +experience, who was acquainted with many plans and devices, and I had +changed my youthful habit of considering things geometrically into a +kind of mechanical mode of view. I felt, intuitively as it were, how +strains and stresses were distributed in any mechanical arrangement, +a faculty also met with in experienced mechanicians and machine +constructors. But I had the advantage over them of being able to make +complicated and specially important relations perspicuous, by means of +theoretical analysis.</p> + +<p>I have also been in a position to solve several mathematical physical +problems, and some, indeed, on which the great mathematicians, since +the time of Euler, had in vain occupied themselves; for example, +questions as to vortex motion and the discontinuity of motion in +<span class="pagenum" id="Page_282">[Pg 282]</span> +liquids, the question as to the motion of sound at the open ends of +organ pipes, &c. &c. But the pride which I might have felt about +the final result in these cases was considerably lowered by my +consciousness that I had only succeeded in solving such problems +after many devious ways, by the gradually increasing generalisation +of favourable examples, and by a series of fortunate guesses. I had +to compare myself with an Alpine climber, who, not knowing the way, +ascends slowly and with toil, and is often compelled to retrace his +steps because his progress is stopped; sometimes by reasoning, and +sometimes by accident, he hits upon traces of a fresh path, which again +leads him a little further; and finally, when he has reached the goal, +he finds to his annoyance a royal road on which he might have ridden +up if he had been clever enough to find the right starting-point at +the outset. In my memoirs I have, of course, not given the reader an +account of my wanderings, but I have described the beaten path on which +he can now reach the summit without trouble.</p> + +<p>There are many people of narrow views, who greatly admire themselves, +if once in a way, they have had a happy idea, or believe they have +had one. An investigator, or an artist, who is continually having a +great number of happy ideas, is undoubtedly a privileged being, and is +<span class="pagenum" id="Page_283">[Pg 283]</span> +recognised as a benefactor of humanity. But who can count or measure +such mental flashes? Who can follow the hidden tracts by which +conceptions are connected?</p> + +<ul class="index"> +<li class="isub6">That which man had never known,</li> +<li class="isub6">Or had not thought out,</li> +<li class="isub6">Through the labyrinth of mind</li> +<li class="isub6">Wanders in the night.</li> +</ul> + +<p>I must say that those regions, in which we have not to rely on lucky +accidents and ideas, have always been most agreeable to me, as fields +of work.</p> + +<p>But, as I have often been in the unpleasant position of having to +wait for lucky ideas, I have had some experience as to when and where +they came to me, which will perhaps be useful to others. They often +steal into the line of thought without their importance being at first +understood; then afterwards some accidental circumstance shows how +and under what conditions they have originated; they are present, +otherwise, without our knowing whence they came. In other cases they +occur suddenly, without exertion, like an inspiration. As far as my +experience goes, they never came at the desk or to a tired brain. I +have always so turned my problem about in all directions that I could +see in my mind its turns and complications, and run through them freely +without writing them down. But to reach that stage was not usually +possible without long preliminary work. Then, after the fatigue from +<span class="pagenum" id="Page_284">[Pg 284]</span> +this had passed away, an hour of perfect bodily repose and quiet +comfort was necessary before the good ideas came. They often came +actually in the morning on waking, as expressed in Goethe’s words which +I have quoted, and as Gauss also has remarked.⁠<a id="FNanchor_31_31" href="#Footnote_31_31" class="fnanchor">[31]</a> +But, as I have stated in Heidelberg, they were usually apt to come +when comfortably ascending woody hills in sunny weather. The smallest +quantity of alcoholic drink seemed to frighten them away.</p> + +<p>Such moments of fruitful thought were indeed very delightful, but not +so the reverse, when the redeeming ideas did not come. For weeks or +months I was gnawing at such a question until in my mind I was</p> + +<ul class="index"> +<li class="isub6">Like to a beast upon a barren heath</li> +<li class="isub6">Dragged in a circle by an evil spirit,</li> +<li class="isub6">While all around are pleasant pastures green.</li> +</ul> + +<p>And, lastly, it was often a sharp attack of headache which released me +from this strain, and set me free for other interests.</p> + +<p>I have entered upon still another region to which I was led by +investigation on perception and observation of the senses, namely, +the theory of cognition. Just as a physicist has to examine the +telescope and galvanometer with which he is working; has to get a clear +<span class="pagenum" id="Page_285">[Pg 285]</span> +conception of what he can attain with them, and how they may deceive +him; so, too, it seemed to me necessary to investigate likewise the +capabilities of our power of thought. Here, also, we were concerned +only with a series of questions of fact about which definite answers +could and must be given. We have distinct impressions of the senses, +in consequence of which we know how to act. The success of the +action usually agrees with that which was to have been anticipated, +but sometimes also not, in what are called subjective impressions. +These are all objective facts, the laws regulating which it will be +possible to find. My principal result was that the impressions of the +senses are only signs for the constitution of the external world, the +interpretation of which must be learned by experience. The interest +for questions of the theory of cognition, had been implanted in me in +my youth, when I had often heard my father, who had retained a strong +impression from Fichter’s idealism, dispute with his colleagues who +believed in Kant or Hegel. Hitherto I have had but little reason to be +proud about those investigations. For each one in my favour, I have +had about ten opponents; and I have in particular aroused all the +metaphysicians, even the materialistic ones, and all people of hidden +metaphysical tendencies. But the addresses of the last few days have +revealed a host of friends whom as yet I did not know; so that in this +<span class="pagenum" id="Page_286">[Pg 286]</span> +respect also I am indebted to this festivity for pleasure and for fresh +hope. Philosophy, it is true, has been for nearly three thousand years +the battle-ground for the most violent differences of opinion, and it +is not to be expected that these can be settled in the course of a +single life.</p> + +<p>I have wished to explain to you how the history of my scientific +endeavours and successes, so far as they go, appears when looked at +from my own point of view, and you will perhaps understand that I am +surprised at the universal profusion of praise which you have poured +out upon me. My successes have had primarily this value for my own +estimate of myself, that they furnished a standard of what I might +further attempt; but they have not, I hope, led me to self-admiration. +I have often enough seen how injurious an exaggerated sense of +self-importance may be for a scholar, and hence I have always taken +great care not to fall a prey to this enemy. I well knew that a +rigid self-criticism of my own work and my own capabilities was the +protection and palladium against this fate. But it is only needful +to keep the eyes open for what others can do, and what one cannot do +oneself, to find there is no great danger; and, as regards my own work, +I do not think I have ever corrected the last proof of a memoir without +finding in the course of twenty-four hours a few points which I could +have done better or more carefully. +<span class="pagenum" id="Page_287">[Pg 287]</span></p> + +<p>As regards the thanks which you consider you owe me, I should be unjust +if I said that the good of humanity appeared to me, from the outset, +as the conscious object of my labours. It was, in fact, the special +form of my desire for knowledge which impelled me and determined me, +to employ in scientific research all the time which was not required +by my official duties and by the care for my family. These two +restrictions did not, indeed, require any essential deviation from the +aims I was striving for. My office required me to make myself capable +of delivering lectures in the University; my family, that I should +establish and maintain my reputation as an investigator. The State, +which provided my maintenance, scientific appliances, and a great share +of my free time, had, in my opinion, acquired thereby the right that I +should communicate faithfully and completely to my fellow-citizens, and +in a suitable form, that which I had discovered by its help.</p> + +<p>The writing out of scientific investigations is usually a troublesome +affair; at any rate it has been so to me. Many parts of my memoirs I +have rewritten five or six times, and have changed the order about +until I was fairly satisfied. But the author has a great advantage in +such a careful wording of his work. It compels him to make the severest +criticism of each sentence and each conclusion, more thoroughly even +<span class="pagenum" id="Page_288">[Pg 288]</span> +than the lectures at the University which I have mentioned. I have +never considered an investigation finished until it was formulated in +writing, completely and without any logical deficiencies.</p> + +<p>Those among my friends who were most conversant with the matter +represented to my mind, my conscience as it were. I asked myself +whether they would approve of it. They hovered before me as the +embodiment of the scientific spirit of an ideal humanity, and furnished +me with a standard.</p> + +<p>In the first half of my life, when I had still to work for my external +position, I will not say that, along with a desire for knowledge and a +feeling of duty as servant of the State, higher ethical motives were +not also at work; it was, however, in any case difficult to be certain +of the reality of their existence so long as selfish motives were +still existent. This is, perhaps, the case with all investigators. But +afterwards, when an assured position has been attained, when those who +have no inner impulse towards science may quite cease their labours, a +higher conception of their relation to humanity does influence those +who continue to work. They gradually learn from their own experience +how the thoughts which they have uttered, whether through literature +or through oral instruction, continue to act on their fellow-men, and +possess, as it were, an independent life; how these thoughts, further +<span class="pagenum" id="Page_289">[Pg 289]</span> +worked out by their pupils, acquire a deeper significance and a more +definite form, and, reacting on their originators, furnish them with +fresh instruction. The ideas of an individual, which he himself has +conceived, are of course more closely connected with his mental field +of view than extraneous ones, and he feels more encouragement and +satisfaction when he sees the latter more abundantly developed than the +former. A kind of parental affection for such a mental child ultimately +springs up, which leads him to care and to struggle for the furtherance +of his mental offspring as he does for his real children.</p> + +<p>But, at the same time, the whole intellectual world of civilised +humanity presents itself to him as a continuous and spontaneously +developing whole, the duration of which seems infinite as compared +with that of a single individual. With his small contributions to the +building up of science, he sees that he is in the service of something +everlastingly sacred, with which he is connected by close bands of +affection. His work thereby appears to him more sanctified. Anyone can, +perhaps, apprehend this theoretically, but actual personal experience +is doubtless necessary to develop this idea into a strong feeling.</p> + +<p>The world, which is not apt to believe in ideal motives, calls this +feeling love of fame. But there is a decisive criterion by which both +<span class="pagenum" id="Page_290">[Pg 290]</span> +kinds of sentiment can be discriminated. Ask the question if it is +the same thing to you whether the results of investigation which you +have obtained are recognised as belonging to you or not when there +are no considerations of external advantage bound up with the answer +to this question. The reply to it is easiest in the case of chiefs +of laboratories. The teacher must usually furnish the fundamental +idea of the research as well as a number of proposals for overcoming +experimental difficulties, in which more or less ingenuity comes into +play. All this passes as the work of the student, and ultimately +appears in his name when the research is finished. Who can afterwards +decide what one or the other has done? And how many teachers are there +not who in this respect are devoid of any jealousy?</p> + +<p>Thus, gentlemen, I have been in the happy position that, in freely +following my own inclination, I have been led to researches for which +you praise me, as having been useful and instructive. I am extremely +fortunate that I am praised and honoured by my contemporaries, in so +high a degree, for a course of work which is to me the most interesting +I could pursue. But my contemporaries have afforded me great and +essential help. Apart from the care for my own existence and that of +my family, of which they have relieved me, and apart from the external +means with which they have provided me, I have found in them a standard +<span class="pagenum" id="Page_291">[Pg 291]</span> +of the intellectual capacity of man; and by their sympathy for my work +they have evoked in me a vivid conception of the universal mental life +of humanity which has enabled me to see the value of my own researches +in a higher light. In these circumstances, I can only regard as a free +gift the thanks which you desire to accord to me, given unconditionally +and without counting on any return.</p> + +<p class="center spa2">PRINTED BY<br><span class="fs_120">SPOTTISWOODE AND CO. LTD.,</span><br> +NEW STREET SQUARE<br> LONDON</p> + +<hr class="chap x-ebookmaker-drop"> + +<p class="f120"><b>WORKS BY JOHN TYNDALL,<br> D.C.L. LL.D. F.R.S.</b></p> + +<div class="blockquot no-wrap"> +<hr class="r10"> + +<p class="neg-indent"><span class="f110"><b>FRAGMENTS of SCIENCE</b>:</span> a Series of Detached Essays,<br> +Addresses, and Reviews. 2 vols. crown 8vo. 16<i>s.</i></p> + +<p class="neg-indent"><span class="f110"><b>NEW FRAGMENTS.</b></span> Crown 8vo. 10<i>s.</i> 6<i>d.</i></p> + +<p class="neg-indent"><span class="f110"><b>LECTURES on SOUND.</b></span> With Frontispiece of Fog-Syren,<br> +and 203 other Woodcuts and Diagrams in the Text.<br> Crown 8vo. 10<i>s.</i> 6<i>d.</i></p> + +<p class="neg-indent"><span class="f110"><b>HEAT, a MODE of MOTION.</b></span> With 125 Woodcuts and Diagrams.<br> +Crown 8vo. 12<i>s.</i></p> + +<p class="neg-indent"><span class="f110"><b>LECTURES on LIGHT, DELIVERED in the UNITED STATES</b></span><br> +in 1872 and 1873. With Portrait, Lithographic Plate, and 59 Diagrams.<br> Crown 8vo. 5<i>s.</i></p> + +<p class="neg-indent"><span class="f110"><b>ESSAYS on the FLOATING MATTER of the AIR in RELATION to<br> +PUTREFACTION and INFECTION.</b></span> With 24 Woodcuts.<br>Crown 8vo. 7<i>s.</i> 6<i>d.</i></p> + +<p class="neg-indent"><span class="f110"><b>RESEARCHES on DIAMAGNETISM and MAGNE-CRYSTALLIC ACTION</b>;</span><br> +including the Question of Diamagnetic Polarity.<br> Crown 8vo. 12<i>s.</i></p> + +<p class="neg-indent"><span class="f110"><b>NOTES of a COURSE of NINE LECTURES on LIGHT</b>,</span><br> +delivered at the Royal Institution of Great Britain, 1869.<br> +Crown 8vo. 1<i>s.</i> 6<i>d.</i></p> + +<p class="neg-indent"><span class="f110"><b>NOTES of a COURSE of SEVEN LECTURES on ELECTRICAL<br> +PHENOMENA and THEORIES</b>,</span><br> delivered at the Royal Institution of Great Britain, 1870. Crown 8vo.<br> +1<i>s.</i> 6<i>d.</i></p> + +<p class="neg-indent"><span class="f110"><b>LESSONS in ELECTRICITY at the ROYAL INSTITUTION</b>,</span><br> +1875-1876. With 58 Woodcuts and Diagrams.<br> Crown 8vo. 2<i>s.</i> 6<i>d.</i></p> + +<p class="neg-indent"><span class="f110"><b>THE GLACIERS of the ALPS</b>:</span><br> +being a Narrative of Excursions and Ascents. An Account of<br> +the Origin and Phenomena of Glaciers, and an Exposition of<br> +the Physical Principles to which they are related.<br> With 6 Illustrations.<br> +Crown 8vo. 6<i>s.</i> 6<i>d.</i>net. ‘Silver Library’ Edition, 3<i>s.</i> 6<i>d.</i></p> + +<p class="neg-indent"><span class="f110"><b>HOURS of EXERCISE in the ALPS</b>.</span><br> +With 7 Illustrations. Crown 8vo. 6<i>s.</i> 6<i>d.</i> net.<br> +‘Silver Library’ Edition, 3<i>s.</i> 6<i>d.</i></p> + +<p class="neg-indent spb2"><span class="f110"><b>FARADAY as a DISCOVERER</b>.</span><br> +With 2 Portraits. Fcp. 8vo. 3<i>s.</i> 6<i>d.</i></p> +</div> + +<hr class="chap x-ebookmaker-drop"> + +<p class="f120">LONGMANS, GREEN, & CO.<br> 39 Paternoster Row,<br> +London New York, Bombay, and Calcutta.</p> + +<hr class="chap x-ebookmaker-drop"> + +<div class="footnotes"> +<p class="f150"><b>Footnotes:</b></p> + +<div class="footnote"><p class="no-indent"> +<a id="Footnote_1_1" href="#FNanchor_1_1" class="label">[1]</a> +Carcinoma recti.</p> +</div> + +<div class="footnote"><p class="no-indent"> +<a id="Footnote_2_2" href="#FNanchor_2_2" class="label">[2]</a> +In his book, <i>On the Limits of Philosophy</i>, Mr. W. +Tobias maintains 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 (<i>Kritik der reinen +Vernunft</i>, Introduction, v. 2nd ed. p. 16); that space has three +dimensions (<i>Ibid.</i> part i. sect. i. § 3, p. 41); that only one +straight line is possible between two points (<i>Ibid.</i> part ii. +sect. i. ‘On the Axioms of Intuition’), as axioms which express <i>a +priori</i> 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 <i>a priori</i> on which my +critic insists.</p> +</div> + +<div class="footnote"><p class="no-indent"> +<a id="Footnote_3_3" href="#FNanchor_3_3" class="label">[3]</a> +Gauss, <i>Werke</i>, Bd. IV. p. 215, first published in +<i>Commentationes Sec. Reg. Scientt. Gottengensis recentiores</i>, vol. vi., 1828.</p> +</div> + +<div class="footnote"><p class="no-indent"> +<a id="Footnote_4_4" href="#FNanchor_4_4" class="label">[4]</a> +<i>Saggio di Interpretazione della Geometria Non-Euclidea</i>, Napoli, +1868.—<i>Teoria fondamentale degli Spazii di Curvatura costante, +Annali di Matematica</i>, Ser. II. Tom. II. pp. 232-55. Both have been +translated into French by J. Hoüel, <i>Annales Scientifiques de l’Ecole +Normale</i>, Tom V., 1869.</p> +</div> + +<div class="footnote"><p class="no-indent"> +<a id="Footnote_5_5" href="#FNanchor_5_5" class="label">[5]</a> +<i>Principien der Geometrie</i>, Kasan, 1829-30.</p> +</div> + +<div class="footnote"><p class="no-indent"> +<a id="Footnote_6_6" href="#FNanchor_6_6" class="label">[6]</a> +Ueber die Hypothesen welche der Geometrie zu Grunde liegen, +Habilitationsschrift vom 10 Juni 1854. (<i>Abhandl. der königl. +Gesellsch. zu Göttingen</i>, Bd. XIII.)</p></div> + +<div class="footnote"><p class="no-indent"> +<a id="Footnote_7_7" href="#FNanchor_7_7" class="label">[7]</a> +Helmholtz’s <i>Popular Lectures</i>, Series I. p. 243.</p> +</div> + +<div class="footnote"><p class="no-indent"> +<a id="Footnote_8_8" href="#FNanchor_8_8" class="label">[8]</a> +Ibid. p. 86.</p> +</div> + +<div class="footnote"><p class="no-indent"> +<a id="Footnote_9_9" href="#FNanchor_9_9" class="label">[9]</a> +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.</p> +</div> + +<div class="footnote"><p class="no-indent"> +<a id="Footnote_10_10" href="#FNanchor_10_10" class="label">[10]</a> +They are algebraical expressions compounded from the coefficients of +the various terms in the expression for the square of the distance of +two contiguous points and from their differential quotients.</p> +</div> + +<div class="footnote"><p class="no-indent"> +<a id="Footnote_11_11" href="#FNanchor_11_11" class="label">[11]</a> +As occurs, for instance, in the above-mentioned work of Tobias, pp. 70, etc.</p> +</div> + +<div class="footnote"><p class="no-indent"> +<a id="Footnote_12_12" href="#FNanchor_12_12" class="label">[12]</a> +<i>Teoria fondamentale, &c.</i>, <i>ut sup.</i></p> +</div> + +<div class="footnote"><p class="no-indent"> +<a id="Footnote_13_13" href="#FNanchor_13_13" class="label">[13]</a> +Ueber die Thatsachen die der Geometrie zum Grunde liegen +(<i>Nachrichten von der königl. Ges. d. Wiss. zu Göttingen</i>, Juni 3, 1868).</p> +</div> + +<div class="footnote"><p class="no-indent"> +<a id="Footnote_14_14" href="#FNanchor_14_14" class="label">[14]</a> +‘Untersuchungen über die ganzen homogenen Functionen von <i>n</i> +Differentialen’ (Borchardt’s <i>Journal für Mathematik</i>, Bd. lxx. 3, +71; lxxiii. 3,1); ‘Untersuchung eines Problems der Variationsrechnung’ +(<i>Ibid.</i> Bd. lxxiv.).</p> +</div> + +<div class="footnote"><p class="no-indent"> +<a id="Footnote_15_15" href="#FNanchor_15_15" class="label">[15]</a> +Compare the Appendix at the end of this Lecture.</p> +</div> + +<div class="footnote"><p class="no-indent"> +<a id="Footnote_16_16" href="#FNanchor_16_16" class="label">[16]</a> +The reciprocal of the square of this distance, expressed +in negative quantity, would be the measure of curvature of the +pseudospherical space.</p> +</div> + +<div class="footnote"><p class="no-indent"> +<a id="Footnote_17_17" href="#FNanchor_17_17" class="label">[17]</a> +Helmholtz’s <i>Popular Scientific Lectures</i>, pp. 232-52.</p> +</div> + +<div class="footnote"><p class="no-indent"> +<a id="Footnote_18_18" href="#FNanchor_18_18" class="label">[18]</a> +Dobrowolsky in <i>Graefe’s Archiv für Ophthalmologie</i>, +vol. xviii. part i. pp. 24-92.</p> +</div> + +<div class="footnote"><p class="no-indent"> +<a id="Footnote_19_19" href="#FNanchor_19_19" class="label">[19]</a> +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.</p> +</div> + +<div class="footnote"><p class="no-indent"> +<a id="Footnote_20_20" href="#FNanchor_20_20" class="label">[20]</a> +See Helmholtz’s <i>Popular Lectures</i>, first series, p. 250.</p> +</div> + +<div class="footnote"><p class="no-indent"> +<a id="Footnote_21_21" href="#FNanchor_21_21" class="label">[21]</a> +I disregard here the view that irradiation in the eye depends on 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.</p> +</div> + +<div class="footnote"><p class="no-indent"> +<a id="Footnote_22_22" href="#FNanchor_22_22" class="label">[22]</a> +Conf. E. Brücke, <i>Die Physiologie der Farben für die Zwecke der +Kunstgewerbe</i>. Leipzig, 1866. W. v. Bezold, <i>Die Farbenlehre, ein +Hinblick auf Kunst und Kunstgewerbe</i>. Braunschweig, 1874.</p> +</div> + +<div class="footnote"><p class="no-indent"> +<a id="Footnote_23_23" href="#FNanchor_23_23" class="label">[23]</a> +Cox’s <i>Aryan Mythology</i>, vol. i. 372. Longmans.</p> +</div> + +<div class="footnote"><p class="no-indent"> +<a id="Footnote_24_24" href="#FNanchor_24_24" class="label">[24]</a> +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.</p> +</div> + +<div class="footnote"><p class="no-indent"> +<a id="Footnote_25_25" href="#FNanchor_25_25" class="label">[25]</a> +This calculation would, however, lose its bases if Maxwell’s hypothesis +were confirmed, according to which light depends on electrical and +magnetical oscillations.</p> +</div> + +<div class="footnote"><p class="no-indent"> +<a id="Footnote_26_26" href="#FNanchor_26_26" class="label">[26]</a> +Mr. Zoellner concludes from photometric measurements, which, however, +need confirmation, that Jupiter still possesses a light of its own.</p> +</div> + +<div class="footnote"><p class="no-indent"> +<a id="Footnote_27_27" href="#FNanchor_27_27" class="label">[27]</a> +Or perhaps also to oxygen. The line occurs in the spectrum of +atmospheric air, and according to H. C. Vogel’s observations was +wanting in the spectrum of pure oxygen.</p> +</div> + +<div class="footnote"><p class="no-indent"> +<a id="Footnote_28_28" href="#FNanchor_28_28" class="label">[28]</a> +Xenophon, <i>Memorabil.</i> I. i. 11.</p> +</div> + +<div class="footnote"><p class="no-indent"> +<a id="Footnote_29_29" href="#FNanchor_29_29" class="label">[29]</a> +Arthur Schopenhauer, <i>Von ihm. über ihn von Frauenstadt +und Lindner</i>. Berlin, 1863, p. 653.</p> +</div> + +<div class="footnote"><p class="no-indent"> +<a id="Footnote_30_30" href="#FNanchor_30_30" class="label">[30]</a> +Preface to the German translation of Tyndall’s +<i>Scientific Fragments</i>, p. xxii.</p> +</div> + +<div class="footnote"><p class="no-indent"> +<a id="Footnote_31_31" href="#FNanchor_31_31" class="label">[31]</a> +Gauss, <i>Werke</i>, vol. v. p. 609. ‘The law of induction discovered +Jan. 23, 1835, at 7 <span class="allsmcap">A.M.</span>, before rising.’</p> +</div></div> + +<div class="chapter"> +<div class="transnote bbox spa2"> +<p class="f120 spa1">Transcriber’s Notes:</p> +<hr class="r10"> +<p>Deprecated spellings or ancient words were not corrected.</p> +<p>The illustrations have been moved so that they do not break up + paragraphs and so that they are next to the text they illustrate.</p> +<p>Typographical and punctuation errors have been silently corrected.</p> +<p>The list of works by John Tyndall has been moved from the front to the + end of the book.</p> +</div></div> + +<div style='text-align:center'>*** END OF THE PROJECT GUTENBERG EBOOK 77725 ***</div> + </body> +</html> + diff --git a/77725-h/images/cbr-3.jpg b/77725-h/images/cbr-3.jpg Binary files differnew file mode 100644 index 0000000..e3281eb --- /dev/null +++ b/77725-h/images/cbr-3.jpg diff --git a/77725-h/images/cover.jpg b/77725-h/images/cover.jpg Binary files differnew file mode 100644 index 0000000..29641c0 --- /dev/null +++ b/77725-h/images/cover.jpg diff --git a/77725-h/images/i_042a.jpg b/77725-h/images/i_042a.jpg Binary files differnew file mode 100644 index 0000000..2aea7c9 --- /dev/null +++ b/77725-h/images/i_042a.jpg diff --git a/77725-h/images/i_042b.jpg b/77725-h/images/i_042b.jpg Binary files differnew file mode 100644 index 0000000..dc01b54 --- /dev/null +++ b/77725-h/images/i_042b.jpg diff --git a/77725-h/images/i_102a.jpg b/77725-h/images/i_102a.jpg Binary files differnew file mode 100644 index 0000000..67d73c5 --- 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