Part 2

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.