Chapter I: Preface (1)

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.