The Analysis of Matter

Newton, "force" plays a great part, and there seems no doubt that he regarded it as a vera causa. If there was action at a distance, the use of the words "central forces" seemed to make it somehow more "intelligible." But gradually it was increasingly realized that "force" is merely a connecting link between configurations and accelerations; that, in fact, causal laws of the sort leading to differential equations are what we need, and that "force" is by no means necessary for the enunciation of such laws. Kirchoff and Mach developed a mechanics which dispensed with "force," and Hertz perfected their views in a treatise[5] comparable to Euclid from the point of view of logical beauty, leading to the result that there is only one law of motion, to the effect that, in a certain defined sense, every particle describes a geodesic. Although the whole of this development involved no essential departure from Newton, it paved the way for relativity dynamics, and provided much of the necessary mathematical apparatus, particularly in the use of the principle of least action. The first physical theory to be developed on lines definitely different from those of Newtonian astronomy was the undulatory theory of light. Not that there was anything to contradict Newton, but that the framework of ideas was different. Transmission through a medium had been made fashionable by Descartes, and unfashionable by the Newtonians; in the case of the transmission of light it was found necessary to revert to the older point of view. Moreover, the æther was never so comfortably material as "gross" matter. It could vibrate, but it did not seem to consist of little bits[Pg 20] each with its own individuality, or to be subject to any discoverable molar motions. No one knew whether it was a jelly or a gas. Its properties could not be inferred from those of billiard balls, but were merely those demanded by its functions. In fact, like a painfully good boy, it only did what it was told, and might therefore be expected to die young. A more serious change was introduced by Faraday and Maxwell. Light had never been treated on the analogy of gravitation, but electricity appeared to consist of central forces varying inversely as the square of the distance, and was therefore confidently fitted into the Newtonian scheme. Faraday experimentally and Maxwell theoretically displayed the inadequacy of this view; Maxwell, moreover, demonstrated the identity of light and electromagnetism. The æther required for the two kinds of phenomena was therefore the same, which gave it a much better claim to be supposed to exist. Maxwell's proof, it is true, was not conclusive, but it was made so by Hertz when he produced electromagnetic waves artificially and studied their properties experimentally. It thus became clear that Maxwell's equations, which contained practically the whole of his system, must take their place beside the law of gravitation as affording the mathematical formula for a vast range of phenomena. The concepts required for these equations were, at first, not definitely contradictory to the Newtonian dynamics; but by the help of subsequent experimental results contradictions emerged which were only removed by the theory of relativity. Of this, however, we shall speak in a later chapter. Another breach in the orthodox system, of which the importance has only become fully manifest since the publication of the general theory of relativity, was the invention of non-Euclidean geometry. In the work of Lobatchevsky and Bolyai, although the philosophical challenge to Euclid was already complete, and the consequent argument against Kant's[Pg 21] transcendental æsthetic very powerful, there were not yet, at least obviously, the far-reaching physical implications of Riemann's inaugural dissertation "Ueber die Hypothesen, welche der Geometric zu Grunde liegen." A few words on this topic are unavoidable at this stage, although the full discussion will come later. One broad result of non-Euclidean geometry, even in its earliest form, was that the geometry of actual space is, at least in part, an empirical study, not a branch of pure mathematics. It may be said that empiricists, such as J. S. Mill, always based geometry upon empirical observation. But they did the same with arithmetic, in which they were certainly mistaken. No one before the non-Euclideans perceived that arithmetic and geometry stand on a quite different footing, the former being continuous with pure logic and independent of experience, the latter being continuous with physics and dependent upon physical data. Geometry can, it is true, be still studied as a branch of pure mathematics, but it is then hypothetical, and cannot claim that its initial hypotheses (which replace the axioms) are true in fact, since this is a question outside the scope of pure mathematics. The geometry which is required by the engineer or the astronomer is not a branch of pure mathematics, but a branch of physics. Indeed, in the hands of Einstein geometry has become identical with the whole of the general part of theoretical physics: the two are united in the general theory of relativity. Riemann, who was logically the immediate predecessor of Einstein, brought in a new idea of which the importance was not perceived for half a century. He considered that geometry ought to start from the infinitesimal, and depend upon integration for statements about finite lengths, areas, or volumes. This requires, inter alia, the replacement of the straight line by the geodesic: the latter has a definition depending upon infinitesimal distances, while the former has not. The traditional[Pg 22] view was that, while the length of a curve could, in general, only be defined by integration, the length of the straight line between two points could be defined as a whole, not as the limit of a sum of little bits. Riemann's view was that a straight line does not differ from a curve in this respect. Moreover, measurement, being performed by means of bodies, is a physical operation, and its results depend for their interpretation upon the laws of physics. This point of view has turned out to be of very great importance. Its scope has been extended by the theory of relativity, but in essence it is to be found in Riemann's dissertation. Riemann's work, as well as that of Faraday and Maxwell, belongs, like the theory of relativity, to the development of the view of the physical world as a continuous medium, which has, from the earliest times, contested the mastery with the atomic view. Just as Newton caused absolute space and time to be embedded in the technique of dynamics, so Pythagoras caused spatial atomism to be embedded in the technique of geometry. Ever since Greek times, those who did not believe in the reality of "points" were faced with the difficulty that a geometry based on points works, while no other way of starting geometry was known. This difficulty, as Dr Whitehead has shown, exists no longer. It is now possible, as we shall see at a later stage, to interpret geometry and physics with material all of which is of a finite size—it is even possible to demand that none of the material shall be smaller than an assigned finite size. The fact that this hypothesis can be reconciled with mathematical continuity is a novel discovery of considerable importance; until recently, atomism and continuity appeared incompatible. There are, however, forms of atomism which have not hitherto been found easy to reconcile with continuity; and, as it happens, there is powerful experimental evidence in their favour. Just at the moment when Maxwell, supplemented by Hertz, appeared to have reduced everything to continuity, the new evidence for an atomic view of Nature began to accumulate. There is still an unreconciled conflict, one set of facts pointing in one direction, and another in another; but it is legitimate to hope that the conflict will be resolved before long modern atomism, however, demands a new chapter. FOOTNOTES: [5] Prinzipien der Mechanik. [Pg 24] CHAPTER III ELECTRONS AND PROTONS PHYSICS, at the present time, is divisible into two parts, the one dealing with the propagation of energy in matter or in regions where there is no matter, the other with the interchanges of energy between these regions and matter. The former is found to require continuity, the latter discontinuity. But before considering this apparent conflict, it will be advisable to deal in outline with the discontinuous characteristics of matter and energy as they appear in the theory of quanta and in the structure of atoms. It is necessary, however, for philosophical purposes, to deal only with the most general aspects of modern theories, since the subject is developing rapidly, and any statement runs a risk of being out of date before it can be printed. The topics considered in this chapter and the next have been treated in an entirely new way by the theory initiated by Heisenberg in 1925. I shall, however, postpone the consideration of this theory until after that of the Rutherford-Bohr atom and the theory of quanta connected with it. It appears that both matter and electricity are concentrated exclusively in certain finite units, called electrons and protons. It is possible that the helium nucleus may be a third independent unit, but this seems improbable.[6] The net positive charge of a helium nucleus is double-that of a proton, and its mass is slightly less than four times that of a proton. These facts are explicable (including the slight deficiency of mass) if the[Pg 25] helium nucleus consists of four protons and two electrons; otherwise, they seem an almost incredible coincidence. We may therefore assume that electrons and protons are the sole constituents of matter; if it should turn out that the helium nucleus must be added, that would make little difference to the philosophical analysis of matter, which is our task in this volume. Protons all have the same mass and the same amount of positive electricity. Electrons all have the same mass, about of the mass of a proton. The amount of negative electricity on an electron is always the same, and is such as to balance exactly the amount on a proton, so that one electron and one proton together constitute an electrically neutral system. An atom consists, when unelectrified, of a nucleus surrounded by planetary electrons: the number of these electrons is the atomic number of the element concerned. The nucleus consists of protons and electrons: the number of the former is the atomic weight of the element, the number of the latter is such as to make the whole electrically neutral—i.e. it is the difference between the number of protons in the nucleus and the number of planetary electrons. Every item in this complicated structure is supposed, at normal times, to be engaged in motions which result, on Newtonian principles (modified slightly by relativity considerations), from the attractions between electrons and protons and the repulsions between protons and protons as well as between electrons and electrons. But of all the motions which should be possible on the analogy of the solar system, it is held that only an infinitesimal proportion are in fact possible; this depends upon the theory of quanta, in ways which we shall consider later. The calculation of the orbits of planetary electrons, on Newtonian principles, is only possible in the two simplest cases: that of hydrogen, which consists (when unelectrified) of one proton and one electron; and that of positively electrified[Pg 26] helium, which has lost one, but not both, of its planetary electrons. In these two cases the mathematical theory is practically complete. In all other cases which actually occur, although the mathematics required is of a sort which has been investigated ever since the time of Newton, it is impossible to obtain exact solutions, or even good approximations. The case is still worse as regards nuclei. The nucleus of hydrogen is a single proton, but that of the next element, helium, is held to consist of four protons and two electrons. The combination must be extraordinarily stable, both because no known