the first magnitude and an immense stride in the direction of the unification of natural causes. But it did not satisfy the thoroughgoing dynamical prejudices of Lord Kelvin, who insisted to the end of his life that he did not "understand" the electromagnetic theory and that it "has not helped us hitherto." Maxwell himself was scarcely less desirous of finding a dynamical foundation for his theory. In fact, its first form was a detailed mechanical model of vortices and idle wheels; in the final form details were avoided by the use of the generalized dynamics of Lagrange and Hamilton, and Maxwell succeeded in showing that certain parts at least of his theory could be based upon dynamical principles. This use of Lagrangian and Hamiltonian methods in the investigation of physical phenomena was a new weapon in the hands of those who sought to reduce them all to a dynamical basis. It has been used with effect by J. J. Thomson, Larmor, and (in application to statistical mechanics) by Gibbs. It makes feasible the ultimate refinement and completeness of dynamical explanation; in place of the potential energy in the Lagrangian function we may substitute the kinetic energy of concealed motions and thus the last vestige of unexplained distance-forces may be swept away. The most thoroughgoing and successful example of this method is the very comprehensive theory of the physical universe contained in Larmor's "Ether and Matter" published in the last year of the nineteenth century. His ether is identical with MacCullagh's rotationally elastic medium; it has imbedded in it centers of rotational strain (the electrons), out of which the atoms of matter may be built up. The only assumptions are that the positive and negative electrons are somehow prevented from destroying each other and that they, with their fields of strain, are capable of motion through the fixed medium. From Hamilton's principle, the Maxwellian equations for the free ether are deduced and, in the presence of matter (electrons), whether at rest or in motion, the same relations hold as those found experimentally. The rotational elasticity of the medium may be produced gyrostatically, so that the potential energy may, if one chooses, be replaced by kinetic. It is interesting to observe that the position, velocity and momentum of a material particle, in this theory, are really Lagrangian, generalized values. The motion of the centers of strain (e. g. in a straight line) cause a slight twisting and untwisting motion of the ether where the true mass and momentum reside. Thus the apparent mass of Larmor's electron varies with its speed as that of cathode rays was afterward found to do; but its dynamical orthodoxy is as sound as that of a steam-engine governor, whose moment of inertia varies with its angular velocity. Notwithstanding the triumphs of the dynamical school of thought, its assumptions and methods were subjected to searching criticism on philosophical grounds particularly by Kirchhoff and Mach. In Kirchhoff's "Lectures on Mechanics," published in 1876, he explicitly renounces the attempt to find the causes of natural phenomena or to "explain " them in the traditional sense; the purpose of mechanics itself (to say nothing of the parts of physics more remote from common observation) is simply the description of phenomena. Forces as causes of motion are rejected; they are merely convenient abbreviations for certain functions of observed motions. In the first lecture he points out that Newton by no means discovered that the force of gravitation was the cause of the motion of the planets which Kepler had described; he only showed that the description was simpler and briefer if expressed in terms of the second differential coefficients instead of the first. Similar ideas have been developed with greater generality by Mach, who finds the final purpose of scientific theories to be economy of thought, and classes the search for causes, explanations and dynamical theories, among the metaphysical prejudices which hinder the progress of science. The criticism of Kirchhoff and Mach is logical and convincing. No unprejudiced person can doubt that, after a discovery is made, it may be interpreted in their way and that, on the whole, this interpretation is the cleanest, most rational and most free from human weakness. But from the pragmatic point of view, and in the light of experience of the course of science in the past, it may well be doubted if their attitude of mind is a useful one in the work of investigation and discovery as distinguished from subsequent criticism and clarification. A somewhat extreme example of militant advocacy of the descriptive method was furnished about twenty years ago by the school of energetics under the leadership of Ostwald. Any use of atomic hypotheses was by them regarded as evidence of feebleness of intellect and slavery to metaphysical prejudices. Their opinions were based upon an incomplete acquaintance with the state of physical knowledge even at that time; they were vigorously opposed in numerous papers by Boltzmann who demonstrated "the indispensability of atomistics in natural philosophy" in a most convincing manner. As we all know, the progress of experimental discovery has long since convinced the energeticians that no adequate description of material phenomena can be given without the use of atomic theories. Boltzmann has also pointed out that even the most elaborate and detailed mechanical theories of Kelvin or Maxwell, for example, are regarded by their authors themselves merely as models; that description by means of models, if accurate and convenient, is quite as legitimate as description by means of differential equations; and that the method could be thus amply justified even on the most sophisticated philosophical principles. It may, I think, safely be said that the most remarkable example in physical science of the purely descriptive theory-the one with the least taint of the fallacy of cause and effect-is Einstein's theory of relativity. All of us who studied our Maxwell in the early nineties or previous to that time, and who have kept an interested eye upon the progress of electrodynamics in the intervening years, are aware of the great difficulties which were encountered in the attempt to extend the Maxwellian electrodynamics to moving bodies. Maxwell and Hertz both went astray in that portion of the subject. We all remember how these difficulties were slowly cleared up, step by step, especially by the masterly work of Lorentz, but with important contributions by J. J. Thomson, Heaviside, Larmor, FitzGerald, Max Abraham and others. What we now call the electron theory had its origin in this attack upon the electrodynamics of moving matter, and was not the result of any prevision that within a few years we should be able to handle, and experiment with, the disembodied electrons themselves. The final puzzle was the reconciliation of the result of the Michelson-Morley experiment with the facts of aberration, the Fresnel "coefficient of entrainment" and other optical knowledge. Most of us can remember the great perplexity which this caused; and it "Populäre Schriften," p. 1. did not at first sight appear to be helped very much by FitzGerald's suggestion, contained in a memoir by Lodge "that the cohesive force between molecules, and, therefore, the size of bodies, may be a function of their direction of motion through the ether; and accordingly that the length and breadth of Michelson's stonesupporting block were differently affected, in what happened to be, either accidentally or for some unknown reason, a compensatory manner. This seemed a rather desperate dodge; and the impression was not removed until Lorentz (who had independently made the same suggestion) showed that just the right alteration of dimensions would take place if the intermolecular forces were of electrical origin. Later the experimental results of Rayleigh and of Brace forced him to the conclusion that the electron itself must be similarly contracted, and one of the consequences of his hypothesis was brilliantly verified by Buch erer. Upon one who had followed step by step this slow and laborious, but highly interesting, course of development, with its constant action and reaction of theory and experiment upon each other, the impression of directness and simplicity made by Einstein's papers of 1905 can scarcely be exaggerated. The difficult and (at first sight) irreconcilable results of experiment, which the older theory had conscientiously "explained," were taken by Einstein as his postulates. There remained only to describe the world as it appears to an observer limited by these restrictive postulates; this proved to be (for Einstein) an apparently easy task and resulted in the Lorentz equations for bodies in motion, slightly improved, in that some relations which Lorentz had obtained only approxi 6 Phil. Trans. R. S., 184, p. 749 (1893). 7Versuch einer Theorie,'' etc., § 92. mately were now exact. Since description and not mechanism is the essence of the method, it is unnecessary to postulate an ether; and since an observer at rest with reference to the ether would have no detectable advantage over one who was in motion, the assumption of an ether was not only useless, but actually in the way of clear description. This rejection of the ether has made Einstein's theory unpalatable to many physicists, while others (as) well as many mathematicians) have been so carried away with its beauty and elegance that the use of the word ether is to them distinctly offensive. A simple rule, however, enables one to converse peaceably with either group separately; the same statements and arguments may be addressed to both, provided the word "observer" is substituted for "ether," or vice versa. If we consider Einstein's theory from the pragmatic point of view we cannot fail to recognize that no new discoveries in electrodynamics have resulted from its suggestions. In this fact there appears to be support for the opinion that a theory of this type is not valuable as an instrument of research, but finds its proper place as a succinct summary of a body of knowledge after that knowledge has been acquired by other means. There are a number of considerations, however, which serve warning against this generalization, of which I will mention but two. as a I would first call your attention to the fact that the development of thermodynamics, as based upon the two empirical laws, exemplifies a method which is very similar to that of Einstein; and we must all recognize its enormous services in the advancement of science. It has constantly served as the guide in important experimental investigations, and has predicted results which could scarcely have been foreseen on the basis of the more detailed molecular and statistical theories. The converse is also true, as Boltzmann so stoutly maintained; and I think we must recognize that the progress of thermodynamics has been greatly facilitated by the interplay and mutual reaction of both types of theory. The second example is a more direct one; it is the remarkable theory of gravitation in which the highly individual genius of Einstein has again manifested itself. It is too early to come to a definite conclusion as to its validity. It has had one striking verification in the deduction of the correct value for the unexplained motion of Mercury's perihelion; but this agreement may conceivably be due to accident and, in any case, its evidence is too slender to be regarded as establishing the theory. But we must face the distinct possibility of its ultimate success; and, in that case, we can not fail to recognize it as a brilliant triumph of the descriptive method. It is difficult to believe that any living physicist except Einstein could have constructed this theory even with the help of Minkowski's highly simplified method of description by means of four-dimensional geometry; but it is quite beyond belief that such a theory could have arisen at the present time by the use of any of the more usual methods of theoretical physics. There is one further matter in this connection to which I should like to invite your attention. It is the question of the complete validity of Einstein's original postulate of relativity. There can be little doubt of its correctness when applied to motions of translation; speaking in terms of the ether, we may be reasonably confident that it is impossible to detect the effects of uniform translation relative to the ether. But little has been accomplished in extending the theory to motions of rotation; indeed, rotation has always been a stumbling-block to a purely relative theory of motion, as soon as dynamical considera tions are introduced. As Maxwell says:8 So far as regards the geometrical configuration of the earth and the heavenly bodies, it is evidently all the same "Whether the sun predominant in heaven Rise on the earth, or earth rise on the sun; But, as we all know, the plane of Foucault's pendulum remains fixed with reference to the stars, and this has usually been interpreted as proving by dynamical means the absolute rotation of the earth. The thoroughgoing relativist replies, however, that the contrary supposition is equally possible; it would merely require a restatement of the principles of mechanics which happen (for some unknown reason) to take on their simplest form when referred to axes fixed with respect to the stars. The new statement of the laws of motion would seem to us very unnatural, but the essential point is not their strangeness, but that they would be different. To cause them to transform into themselves, as Maxwell's equations do when subjected to Lorentz-Einstein transformation, would apparently require curious assumptions of curved space, and of time recurrent after twenty-four-hour periods, which would certainly be very foreign to the ordinary habits and preferences of the human mind, the whether we assume that these habits are inherent or acquired. Even from the point of view of convenient description it seems likely that we shall do better by adhering to the belief that the stars are fixed and that the earth rotates. We must, however, 8"Matter and Motion," p. 154 (Van Nostrand, 1878). 9"Paradise Lost," Book 8, 11. 160 et seq." admit that relativists are quite within their rights when they demand an answer to the question, "Fixed with reference to what; rotates relative to what?" Here, it seems to me, is a possible field of usefulness for the ether in addition to its original function of serving as nominative case to the verb "to undulate." This appears the more likely when we consider that the earth's magnetism has never received an explanation or, if one chooses, a description which connects it with other physical phenomena. I have left to the end the consideration of the most revolutionary change which the twentieth century has brought about in the outlook and methods of theoretical physics-the rapid development and great successes of the quantum hypothesis of Planck. As we have seen, the fifty years following the discovery of the conservation of energy were marked by the steady progress of dynamical theories and the conquest by them of one disputed position after another. It is true that the victory was never quite complete, that the models were always in some degree imperfect and approximate; but the success was, on the whole so great that it seemed to justify the hope that only time and labor were necessary to clear away present difficulties as so many had been overcome in the past. It had not been easy to bring thermodynamics and irreversible processes into the dynamical system, but so far as material systems. were concerned, most physicists were in agreement that it had been successfully done. It is true that a violation of the second law of thermodynamics could not be shown to be impossible; but its improbabil ity was so great that there was no reasonable expectation of its ever being observed by finite human beings. The most complete and general exposition of this great triumph of the dynamical hypothesis is contained in the "Statistical Mechanics" of Willard Gibbs, which was published in 1902, but which had been completed and given in the form of academic lectures by the author for some years previous to that date. As in all of Gibbs's work the assumptions and the results were of a very general character; but he was quite aware that at one point they were too restricted. He says:10 Although our only assumption is that we are considering conservative systems of a finite number of degrees of freedom, it would seem that this is assuming far too much, so far as the bodies of nature are concerned. The phenomena of radiant heat, which certainly should not be neglected in any complete system of thermodynamics, and the electrical phenomena associated with the combination of atoms, seem to show that the hypothesis of systems of a finite number of degrees of freedom is inadequate for the explanation of the properties of bodies. The difficulties involved in the possession by the continuous ether of an infinite number of degrees of freedom were brought more clearly to light in 1900 by Lord Rayleigh's formula for black body radiation. It was quite irreconcilable with the measurements of Paschen and, moreover, it led to a kind of superdissipation of energy into high frequency vibrations of the ether which appeared entirely out of accord with the facts of empirical thermodynamics. Paschen's observations were well represented by the formula which had been obtained by Wien, who assumed the Maxwellian distribution of velocities among the molecules of the black radiator, and also that the wave-length radiated by any molecule was a function of its velocity. Later experiments by Lummer and Pringsheim and by Rubens and Kurlbaum, with longer wave-lengths and higher temperatures, approximated to the Rayleigh formula. 10 Statistical Mechanics,'' p. 167. |