INERTIA A N D COERCION .
BY LADISLAS NATANSON
If we cast a glance upon the universe as we see it to be, we shall be confronted with a n infinite number of events, an endless variety of transformations, so that nature would at first sight look only like a confused jumble of innumerable qualities. Whether this be really so, or whether any qualitative differences at all underlie appearances, is a problem that (at the present time) we must confess ourselves unable not only to solve, but even to examine properly. However worthy of respect the various branches of science may be as a display of man’s mental powers, in the full light of nature’s splendour, they appear to be merely uncertain gropings : as indeed they are. I. Accustomed as we are i n natural philosophy to consider no other object but inanimate nature, we have hitherto been unable to take any other standpoint. Kot only so, but we are not yet enabled to survey with a single glance the whole of the horizon thus limited by our own standpoint; as yet we have not succeeded in combining the accuniulated knowledge concerning inanimate nature into a homogeneous system. We can say no more than that far as we may be from the attainment of such a goal we still are aiming to attain it. 11. T h e phenomena of inanimate nature may be divided into two broad categories, those which bear a character of permanence, and those which tend to subside. We know, for instance, that the rotational and translational motion of the earth is a permanent phenomenon, like that of the moon, of the planets and comets, of the suns and planetary systems, as they move through space. T h e movement of terrestrial bodies is not less permanent, so long as nothing occurs to Lecture delivered a t the annual meeting of the Cracow -4cademy of Sciences, by Dr. Ladislas Natanson, Professor of Natural Philosophy in the Yagellonian University, Cracow.
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interfere with it. Permanence i n such phenomena is called inertia. I n the principle of inertia, science teaches us a most important lesson in the economy of nature, though it may be no more than a first instalment, commanding a particular case, and perhaps an approximation. Radiation affords us another example of permanence; I mean those periodic, rhythmical disturbances which are perpetually rolling through boundless, all-pervading aether. T h i s is a fundamental phenomenon, and is, it may be, the most fundamental of all. 111. There are a great many natural phenomena of which the course runs on quite different lines. If a n iron rod be heated on one end and cooled a t the other, heat,’ in popular language, will ‘flow ’ from the hotter to the colder portion, T h e motion of heat, intense at the outset, becomes sluggish ere long, and a t last disappears entirely ; we reach at last a state of uniform temperature, or equilibrium. Various phenomena follow similar laws. I n a similar manner two gases diffuse into each other, sugar diffuses i n water, and gold in lead. A current of air in a closed room subsides in the same way. This is also the case with thousands of chetnical reactions. An electric disturbance in any kind of matter tends to disappear, like the non-uniformity of temperature in an iron rod. In these and i n many other similar cases we are confronted with something quite the opposite to inertia ; we see a continuous succession of variable and transient states which tend to subside (ever more and more sluggishly, it is true), to suppress the original disturbance, to bring about equilibrium. IV. Reverting now to the phenomena of inertia : of course they are mere abstractions, isolated from reality. Motion, for instance, is what dynamics investigates. But in nature there is no such thing as pure motion. Every real movement is intricately connected with other phenomena, either of the first category or of the second. Dynamics treats therefore of ideal phenomena ; its very foundations are wanting i n breadth. T h a t is a thing understood by every scientific man ; and yet we culti-
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grate dynamics and not infrequently set it up as a standard. For we cannot do otherwise. Our aim is to grasp nature; and although nature is more complex than toiigue can tell we desire it all the same. True, if we place ourselves a t the standpoint of dynamics, we stand committed to a one-sided view, being thus induced to set aside quite a world of real relations; but then we at any rate can understand something of nature. T h e way was shown by the master mind of Newton. He was the first to show that the motion of celestial bodies is ruled by laws which the mind of man can grasp; and by a generalization from this great achievement he discovered the universal laws of motion and thus created abstract dynamics. As we reflect upon this at the present day, we cannot help feeling perplexed. Take our solar system, for example. Prom the inimense, coherent, and unique totum of phenomena with which that systeni confronts us, no constituent part can be separated except by a process of the nature of a fiction. How could Newton investigate gravitation without inquiring, for instance, whether the sun’s heat is constant, or what changes occur in terrestrial magnetism, or what chemical reactions take place in the system ? Why should electromagnetic phenomena have, a priovi, no influence on the motion of the various parts of the system? Again, why should not the motion of the earth, for instance, react upon electromagnetic phenomena upon its surface ? For almost a century physicists have sought to discover an effect of the earth’s motion upon optjcal phenomena which they observe i n their laboratories and observatories. We must change, therefore, and to some extent reverse, the position we have taken. It is, we must say, a fact that dynamics can exist as a separate. self-consistent science ; the theory of pure optics, the electromagnetic theory, and other sciences, are likewise, to a certain extent, independent facts. Obviously, in the particular case on which we are engaged and to which our endeavours must conforni, the problem of nature resolves itself, probably approximately, into a number of independent problems. V. Such is the origin of every science. They have all arisen from a possibility of distinguishing, i n the complexity of
nature, one particular and relatively simple problem to be solved. To distinguish this is necessary at the outset of any new science ; it is useful so long as that science grows in certainty and in power. Nevertheless, in presence of the close concatenation and unity which we see in nature, it is but an artifice inconipatible with her intrinsic harmony. And, with the development of science, this incompatibility becomes evident ; the mind of man struggles to pull down the barriers which itself had raised; a tendency arises to alter the aiin of the science, to broaden its foundations and elevate its standpoint. We say, for instance, that we observe in inanimate nature dynaniical, thermal, chemical, electromagnetic phenomena. But we must not forget that all this is merely conventional language ; these phenomena are abstractions, not real phenomena, nor even parts of real phenomena; rather, so to speak, sections through phenomena. As a n architect represents a building by means of vertical and horizontal projection, so does tlie natural philosopher in his manifold theories ; he gives 11s sections through nature, viewed from different standpoints. VI. Thermodynamics, in a sense, is an attempt to do away with restiicted standpoints. It claims not to divide problems in order to investigate them. In tlie first place, thennodynamics shows that ordinary dynamics refers to one particular case only ; and that infinitely many other c a ~ e sof dynatnics are possible. When, for instance, Newton and his followers i n the XVIII. century endeavoured to discover the law of propagation of sound by ordinary dynamics, they failed. T h e problem belongs to the field of what we should call adiabatic dynamics; and when Laplace solved i t he virtually created a second dynamics by the side of the first ; this, together with numberless others besides, at present forms part of general thermodynamics. There are, however, cases to which no form of dynamics can be made to apply. Such are those in which the two categories of phenomena -those which persist and those which tend to subside - are so connected that we cannot treat them sepa-
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rately without doing them violence, without breaking- essential links which bind them together. And thermodynamics -as we may now define it - is a theory which discards such division of phenomena. VII. This, therefore, is the sin1 of thermodynamics ; it is unquestionably as great and daring as its scope is immense. No wonder then if as yet its achievement is but partial. S o far, we have established a theory of equilibrium. Though thermodynamics has not yet recognized such general laws as may govern the coume of transformations, it has discovered one essential property, common to all those which may possibly take place, the characteristic feature which is borne by phenomena conipatible with nature. Suppose a material systeni in which the conditions imposed are such that only changes deprived of this characteristic -and therefore contrary to nature - can occur. Then our conditional hypothesis and the laws of thermodynamics, taken together, forbid the possibility of any phenomenon occurring in the system, and since no change can take place, equilibrium must prevail. By thermodynamics, therefore, we are able to foresee the state of equilibrium towards which the so-called subsiding ” phenomena tend to approximate. A fluid, e. g., may, if certain conditions are satisfied, be in a state of equilibrium ; thermodynamics, therefore, furnishes us with the foundations of hydrostatics and of the theory of capillarity. I n like manner, and from the same source, the theory of elasticity receives its fundamental basis, Electrostatics and other branches of science may be likewise built up upon a thermodynamical basis. Some time ago the discovery was made that an equilibrium state of radiant energy is possible ; and from this discovery a therniodynamical theory of radiation has sprung up. Ice may be i n equilibrium with water, water with steam, a salt-crystal with its solution, a solution with its solvent : of all these and of innumerable other equilibria the laws have been found by thermodynamics. And by thermodynamics has chemistry been triumphantly reduced to scientific order. VIII. As the limits to its sphere of utility recede, the fundamental ideas which underlie thermodynamics, becoming more ((
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and more general, give us an ever bolder outline of what i n reality is essential. And as this great work (in which the name of J. Willard Gibbs will never be forgotten) continues to advance, we become aware, not without some surprise, that generalized thermodynamics tends more and more to similarity with the time-honoured forms assumed by ordinary statics since more than a hundred years ago. Every statical problem is, as we know, solved by the principle called the theorem of virtual work. T h e contents of statics is set forth here i n one formula, in a single line. Nor should we consider this surprising : this knowledge sums u p all essential statical knowledge accumulated from the times of Arcliiniedes to those of Lagrange. Kow, in the same manner, the therinodynaiiiical theory of equilibria enables us to put its contents into one simple formula ; and that formula is in all respects similar to the corresponding one of statics. For the sake of convenience, we frequently introduce into statical reasoning a quantity which depends upon the state of the system, and is called potential. I n thermodynaniics we also introduce a theriiiodynaniical” potentia! ; and we find it to be as useful as the ordinary potential in statics. Finally, the thermodynamical potential is a mere generalization of the statical one ; it is obtained by the consideration of the thermal aspects of equilibria. Here, therefore, is something beyond a merely formal analogy; here is the union of two sciences, one of them being absorbed into the other. Such is at present the stage reached by the theory of equilibria. As we have remarked, it is but one fragmentary part of thermodynamics as required by natural philosophy. Equilibrium is only a liinit to phenomena ; when we know it, we know but the surface of reality ; we have sailed around an island, but have not gone into the interior. Science never stands still, and the theory of equilibria cannot be an ultimate goal. W e must go on to study the laws which preside over the $mgnm of phenomena. Possibly we may not discover them ; but then our successors shall profit by our failures. IX. Let us return to the phenomena which tend to subside. Fourier, one of the masters of our science, has given in ((
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one particular department - the one which deals with thermal conductibility-a beautiful theory, areal pattern of niatlieniatical reasoning. Other investigators have followed him in other directions, but on similar lines, so that there is now in physics quite a number of Fourier-theories.” These, however, are only a step in the right direction ; nature cannot be completely mastered, even by beautiful theories. Let us see whether we may not venture upon a scheme of generalization. Suppose, in a definite material system, a disturbance the nature of which is known, and which tends to subside. Consider, as the quantitative expression of the progress of the phenomenon, a j h x of a certain quantity, in unit time, across unit surface. TT7e obtain, for instance, a convenient measure of the progress of diffusion of two gases into each other if we know the flux of each of them in every part of the system. T h e unit flux of momentum or of energy affords likewise a convenient nieasiire of the progress of subsidence in the motion of a fluid, or in the motion of heat. But, now, upon what does such a flux depend ? It depends in general, we may say, upon the stimulus ’ of the phenomenon. This, in the case of the diffusion of two gases into each other, is their non-uniform density : the slope or space variation of one density or the other approximately measures the intensity of the stimulus at work. I n the pheiiomena of viscous motion-subsidence, the stimulus is, approximately, the space-variation of velocity. In thermal coilductibility, it is the space-variation of temperature, or nearly so. Fourier asslimes that the flux is always in direct ratio to the stimulus. This is true, but only approximately. T h e activity of the stimulus i s not co?z@en‘ to PYoduciFzg the flux; it tends continually to change its intensity. Hence, the stimulus not only sets in inotion the flowing quantity, it gives impetus to the flux as well. T h e impetus may be extremely feeble, as for instance, in Fourier-phenomena: but the fact that it had been altogether neglected was sufficient to disconnect the Fouriertheories from the remaining chapters of natural philosophy. Why is this impetus so slight? Because it is to a great ex-
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tent thwarted by coercion, a powerful though latent resistance which is always ready to act in material bodies. We do not know what its true essence is; but, Maxwell leading the way, we have found the law of coercion to be quite simple. Coercion, though powerful, is not infinitely powerful, as is implicitly assumed by the Fourier-theories, In the case of the interdiffusion of oxygen and nitrogen, for instance, the resistance of oxygen impedes the diffusion of nitrogen; but if it precluded it altogether, diffusion could not take place at all. Coercion diminishes, but does not destroy, the velocity of the flow. Now flow,in a gas, implies impetus, since a gas has inertia. Heat a h possesses inertia, though the inertia is but slight. T h e inertia of heat when conducted through a gas may be regarded as the inertia of the separate molecules in motion. In a metal that conducts heat the inertia may be set down to the account of electrons; for, whatever electrons may be, they behave as (tiue or apparent) masses. But the important point is to notice the fact of inertia, and understand it in its bearings. X. I n the foregoing we have mentioned only one kind of stimulus, which we may call (intrinsic.’ T h e intrinsic stimulus is simply related to the thermodynamical potential, at least in the simpler cases ; it is equal to the space-variation of this potential. But force, as defined in dynamics, also acts as a stimulus ; this we may call the accidental stimulus. T h e resultant of all the stimuli, of whatever kind, or ultimate stimulus, affords a convenient generalization of force as introduced by ordinar!. dynamics. I n the case of mass-motion, the intrinsic stimulus vanishes, the rate of change of the flux is reduced to ordinary acceleration, and since in pure dynamics there is no coercion, we return, from the laws of flux and stimulus, to Newton’s two first lams of motion. T o sum u p : in pure dynamics, i n abstract hydrodynamics and theory of elasticity, we disregard coercion. I n the Fouriertheories, we take no account of inertia. But science cannot stand contented with approximations. I n every phenomenon in which matter is implied, at once coercion and inertia play a
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part ; and the quantitative ratio they bear to each other may vary to the uttermost. There is only one system from which coercion is totally absent : the vacuum or universal aether. Phenomena which take place in pure aether are the free play of unimpeded inertia. XI. About the middle of the X I X . century we learned of Lord Kelvin to recognize in nature the universal and uninterrupted working of coercion. I t may be that, in our attention to coercion, we have come to underrate inertia - that exuberance, as i t were, of the activity of nature. We have just seen instances in which some inertia is manifested, yet which are usually considered as standard cases of coercion. Whenever a change, a phenomenon, takes place, a stimulus must be active ; and its activity must create some kind of impetus -no matter how small. There is one instance of inertia without coercion ; as already remarked, it is afforded by phenotnena displayed in pure aether. Without inertia there is not any known instance of coercion. These two are the poles on which the world of phenomena revolves ; neither is less essential than the other. XII. Persistent phenomena interfere in a n endless multiplicity of ways with those which subside. If we know the laws of both taken separately, we shall be able to elucidate the manner i n which they conibine, producing intricate compound phenomena. Let us again return to dynamics. NewtOn formulated the laws of motion, but i n a concrete form which bore traces of the way that led to their discovery. After Newton came Lagrange and other great philosophers who extended the scope of his doctrine. I n Newton’s dynamics the independent variables are true space-coordinates. In Lagrangian dynamics we may adopt any independent variables we choose ; the rules of generalized dynamics shall not cease to apply. Now we may express these rules in various ways. If we have chosen variables (it may be, in the most general manner), we write down the Lagrangian equations of motion ; these, in generalized dynamics, are the variability-equations. But we can avoid the very appearance of a choice of variables. I n this case there are no explicit equations of variability, but we have a general method for deducing
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them in every case; this is Hamilton’s principle of varying action. Now, think of the case of a pendulum oscillating in a viscous fluid ; or of a gas in which expansion and heat-conduction occur sinlultaneously ; or of a crystal of salt dissolving as it drifts down a river; or of a lump of ice in which elastic vibrations are propagated while it melts. Such problems lie beyond the proper limits of generalized dynamics ; and thermodynamics, as it is now, cannot solve them. They belong to the province of what has been called Thernzokinetics; a province explored as yet by only a small band of students, with Prof. Duhem at their head. And yet it has attained a remarkable result. Thermokinetics presents us once more with Lagrangian equations and with Hamilton’s principle, but in a slightly generalized and supplemented form. Now it is a most important fact, that the extension which is necessary to transfer the Lagrangian equations and Hamilton’s principle from dynamics to thermokinetics is a t bottom the same which (as we have said) makes of the principle of statics the fundamental proposition of the thermodynamical theory of equilibria. XIII. I n truth, all that we know concerning nature is no more than an attempt. I t is plain that we are only starting on our way. And yet, even as it is, standing upon the threshold of knowledge, we are dazzled at every step we have made forwards, and then it takes time to accustom our eyes to the spectacle which expands before us. What then must be infinite nature ?
