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Part 1. The Mechanical Theory

Lecture 5: Molecular Mechanics: Ideals of Matter

(b) The ideal of matter. The old atomism strictly mechanical but inadequate. Its conversion into one strictly dynamical by Boscovich and the French. The resolution of this in turn into the ‘kinetic theory.’

The nature of the primordial fluid examined: it is made up of negations, and is thus indeterminate: prima materia.

Relation of its mass to the ‘quasi-mass’ of the vortices: the latter becomes a complicated problem. The kinetic ideal in danger from ‘metaphysical quagmires.’ To avoid this impasse it is proposed to make energy fundamental.

Results of inquiry into mechanical theory thus far: Relation of the three sciences, Analytical Mechanics, Molar Mechanics, Molecular Mechanics. The first stands completely aloof from concrete facts. The attempt to apply it to these without reserve leaves us with a scheme of motions and nothing to move.

To molar mechanics belongs the rôle of stripping off the physical characteristics of sensible bodies; to molecular mechanics, the rôle of transforming these characteristics into mechanisms, and the mechanisms into ‘non-natter in motion.’ The mechanical theory as a professed explanation of the world thus over-reaches itself.

As mechanical science has advanced, its true character has become increasingly apparent—its objects are fictions of the understanding, and not conceivably presentable facts.

The kinetic ideal shows this best of all, for some of its upholders dream of ‘replacing’ dynamical laws by kinematical. The refutation the more striking because they imagine they are all the while getting nearer to ‘what actually goes on.’

It is upon an uncritical prepossession of this kind that the mechanical theory has rested all along. Descriptive analogies have been regarded as actual facts; yet are nothing but the inevitable outcome of the endeavour to summarise phenomena in terms of motion. A moral drawn from the Pythagoreans.

But mechanical science has so far failed even to describe facts in its own terms.

WE have found physicists protesting with great vehemence against being saddled with any metaphysical conceptions of matter as a substance underlying phenomena. Yet there is only one of the three chief theories of matter that might possibly clear itself of this stigma, and that is the old atomic theory of Democritus or Lucretius; but this, oddly enough, has always claimed to be a theory of substance. In point of fact it is the most phenomenal of all; for the hard atom, apart from its being absolutely hard, differs from tangible bodies only in respect of size and indivisibility. The collisions of such atoms again are essentially phenomenal, though actually beyond the limits of direct perception. Such collisions too are the very type of that plain, straightforward mechanical action, which alone Galileo, Newton, and Huygens—the founders of modern mechanics—were willing to recognise. You will remember the often-quoted letter of Newton to Bentley, in which he declared it to be “inconceivable that inanimate brute matter should … operate upon and affect other matter without mutual contact.” This then is logically the one genuine and original mechanical theory. But absolute hardness is ideal and transcends experience, whereas for the physicist bodies are real and empirically given. I think we may say that whoever ventures to apply to any real thing such adjectives as ‘absolute’ or ‘infinite’ or ‘perfect’ or ‘simple’—the terms being strictly used—has, however much he may dislike it, embrangled himself with metaphysics. Such at least has been the fate of the Lucretian atom, when defined as absolutely hard. Whether Lord Kelvin's perfect fluid fares any better, we can consider later. But let us first notice some of the antinomies besetting the older ideal atom.

Rigid bodies of sensible dimensions are described as respectively elastic or non-elastic, according as they do or do not resume their original shape after being strained. Absolute rigidity, however, absolutely excludes deformation, hence the hard atom can neither be elastic nor non-elastic. What then will happen when two such atoms collide? The problem is strictly indeterminate, so that—as has been said—as often as such an event occurs, the course of the world is at least as uncertain as an act of the purest free will could make it.1 “Take a series of very inelastic bodies such as butter, lead, etc.,” says P. du Bois-Reymond, “and then a series of very elastic bodies, such as india-rubber, ivory, etc. Of which of these two series is the absolutely hard the limit? Obviously of which we like, or of some mean between both.”2 If we decide to regard the atoms as non-elastic, then, when two collide, we must conclude that kinetic energy disappears without an equivalent amount of potential energy taking its place. If we prefer to regard them as elastic, we are then compelled to infer that their motions are instantaneously reversed, in other words, a finite momentum is produced in no time. And if we combine the two, we combine these consequences; both of which contradict our fundamental axioms. The fact is that rigidity, whether accompanied by much or little elasticity, is not a property of mass as such, but a physical property of matter. But if a physical property, then rigidity has to be explained by dynamical transactions between masses, or the mechanical theory fails to redeem its pledge. In other words, it is not open to the physicist to explain—or, as is now said, to describe —rigidity and elasticity in terms of rigidity and elasticity. The retention or restitution of a given shape or configuration implies mechanical or dynamical relations between masses and has to be accounted for. So by inexorable logic the “many hard, impenetrable particles,” which Newton was content to regard as “primitive,” were resolved step by step into the mass-points or centres of force of Boscovich and the French analysts. But as contact action, i.e. action of the straightforward mechanical type, is impossible between mass points, it was replaced by action at a distance, sometimes attractive, sometimes repulsive, according as the distance or other circumstances might vary. The strictly mechanical theory became in fact strictly dynamical.

A word or two of historical explanation seems called for as to this opposition between two terms—I mean ‘mechanical’ and ‘dynamical,’ which are nowadays often regarded as synonymous. The term ‘mechanical’ however, seems appropriate only to motions produced by immediate displacement, as in machines, to contact action in other words. Newton, who—as we have seen—regarded action at a distance as “ so great an absurdity, that I believe,” he writes to Bentley, “no man who has in philosophical matters a competent faculty of thinking can ever fall into it,” finding himself unable mechanically to explain the working of gravitation, contented himself meanwhile with describing the motions produced. But he began early and persisted long in the attempt to discover some medium and mode of operation such as would enable him to explain gravitation by contact, instead of assuming it to be a force “innate, inherent, and essential to matter.” However his friend and contemporary, the youthful Roger Cotes, though anything but a fool, rushed in where the master feared to tread. In his preface to the Principia, Cotes definitely asserted the doctrine of direct action at a distance, and maintained that gravity is no more an occult property of matter than extension, mobility, or impenetrability; since it was, he held, as plainly indicated by experience as they were. “And when”—I here quote Maxwell—“the Newtonian philosophy gained ground in Europe, it was the opinion of Cotes rather than that of Newton that became most prevalent, till at last Boscovich propounded his theory that matter is a congeries of mathematical points, each endowed with the power of attracting or repelling the others according to fixed laws. In his world, matter is inextended, and contact is impossible. He did not forget, however, to endow his mathematical points with inertia.”3 Thus Newton's position was exactly inverted. The solid primitive particles of various sizes and figures, in which Newton inclined to believe, were rejected; and the inherent forces acting through a vacuum, which he disclaimed as absurd, were accepted as the reality to which all the physical properties of matter were due. This is what I meant by saying that his strictly mechanical theory was transformed into one strictly dynamical.

One step in this transformation seems, as I have said, logically inevitable, the reduction of finite molecules to infinitesimal mass-points. Not so the second—the attribution to such mass-points of intrinsic forces. We have seen already that in abstract mechanics this conception of vires insitœ or substantial forces is rigorously scouted. Force is there a purely relative conception, a name for the rate of change of momentum of one mass referred to the position of other masses in the same “field.” Unless then Boscovich's metaphysical idea of forces inherent in a mass-point can be replaced by the mathematical idea of external forces acting at a point, molecular physics cannot be regarded as merely dynamical in the looser modern sense. Central forces when not used geometrically, as by Newton, i.e. merely to describe observed motions, but metaphysically, to explain action at a distance, are incompatible with modern mechanics. They become part of what Professor Tait calls a “very old but most pernicious heresy, of which much more than traces still exist even among physicists.”4 But it must certainly be allowed that the progress of physics has steadily discredited this usage. Faraday's experimental researches into electricity and magnetism, the resolution of heat into “a mode of motion,” and many other lines of investigation tend to confirm the kinetic ideal of matter, which has been aptly described as the theory that matter is non-matter in motion—the non-matter, being of course, Lord Kelvin's ideal fluid.

It is this kinetic, or perhaps I should say hydrokinetic, ideal of Lord Kelvin and his school, that, so far as I can gather, is the ideal of matter prevalent in the present day among such physicists as venture to stir beyond their equations. Any one with a weakness for Hegelian dialectic might easily discover the famous triadic development of thought in the advance from what was in the main Newton's ideal of matter through the ideal of Boscovich to that of Faraday and later British physicists. There seems to have been complete opposition between Newton's conceptions as to what matter really was and the descriptive apparatus of central forces acting across empty space by which he simplified and extended the more cumbrous apparatus of Kepler. Boscovich's doctrine was thus the precise antithesis of Newton's, for he took Newton's descriptive apparatus for the reality, and discarded his solid, impenetrable particles as false. Boscovich's atoms were strictly mass-points; occupation of space with him was due entirely to substantial forces, not to the absolute hardness of primitive particles; and all strictly mechanical action of the push and press kind was replaced by attractions or repulsions acting at a distance. The kinetic theory can be regarded as a synthesis of these contraries. There is no action at a distance; but then there is no empty space: action and reaction are to be explained, not by impact, but by the physical continuity of the plenum. There are no hard atoms; yet the atom occupies space and is elastic in virtue of its rotatory motion.

Faraday, who has been called a disciple of Boscovich, made the first step on in the course of his wonderful electrical researches. He shewed that in the part of space traversed by magnetic force there exists a peculiar tension; as Maxwell puts it, “that wherever magnetic force exists there is matter”—that is to say, an electromagnetic medium or ether. Again Faraday's discovery of the magnetic rotation of the plane of polarised light, together with Maxwell's identification of the rate at which light and electro-magnetic disturbances are propagated, confirmed as this has been by the crucial experiments of Hertz, makes it reasonable to identify the luminiferous and electro-magnetic media. The second great step towards this new ideal begins with the mathematical investigation of Helmholtz into the properties of vortex motion. Though apparently not suggested by Faraday's work, the two were soon brought into connexion; for Lord Kelvin found that the medium when under the action of magnetic force must be in a state of rotation, that is to say, in Maxwell's words “small portions of the medium, which we may call molecular vortices, are rotating, each on its own axis, the direction of this axis being that of the magnetic force.”5 Finally, Helmholtz's demonstration of the conservation of vortex-motion in a perfect fluid led Lord Kelvin to his famous vortex-atom theory, of which I have already spoken, and which in its main features is known to everybody. According to the kinetic ideal of matter, then, both atoms and ether are resolved into motions of one ultimate fluid, which is defined as having “no other properties than inertia, invariable density, and perfect mobility; and the method by which the motion of this fluid is to be traced is pure mathematical analysis.”6

Let me quote two versions of what is expected of this ideal from two of its most able and hopeful supporters. Dr. Larmor, in a paper in the Royal Society's Proceedings of 1893, writes: “It has been in particular the aim of Lord Kelvin to deduce material phenomena from the play of inertia involved in the motion of a structureless primordial fluid; if this were achieved it would reduce the duality, rather the many-sidedness, of physical phenomena, to a simple unity of scheme; it would be the ultimate simplification.” This brief statement is clear and modest by comparison with the following deliverance of Professor Hicks in his Address to Section A at the last meeting (1895) of the British Association: “While on the one hand,” said Professor Hicks, “the end of scientific investigation is the discovery of laws, on the other, science will have reached its highest goal when it shall have reduced ultimate laws to one or two, the necessity of which lies outside the sphere of our cognition. These ultimate laws—in the domain of physical science at least—will be the dynamical laws of the relations of matter to number, space, and time. The ultimate data will be number, matter, space, and time themselves. When these relations shall be known, all physical phenomena will be a branch of pure mathematics. We shall have done away with the necessity of the conception of potential energy, even if it may still be convenient to retain it; and—if it should be found that all phenomena are manifestations of motion of one single continuous medium—the idea of force will be banished also, and the study of dynamics replaced by the study of the equation of continuity.”

Every sentence in these remarks would repay criticism, if we could spare the time. As it is, I must content myself with an occasional reference in the more general criticism of this ultra-physical ideal to which we may now pass. But first, I will ask your indulgence if I quote part of yet another paragraph from this presidential address. “Before, however, this can be attained,” Professor Hicks continues, “we must have the working drawings of the details of the mechanism we have to deal with. These details lie outside the scope of our bodily senses; we cannot see, or feel, or hear them, and this, not because they are unseeable, but because our senses are too coarse-grained to transmit impressions of them to our mind. The ordinary methods of investigation here fail us; we must proceed by a special method, and make a bridge of communication between the mechanism and our senses by means of hypotheses. By our imagination, experience, intuition we form theories, we deduce the consequences of these theories on phenomena which come within the range of our senses, and reject or modify and try again. It is a slow and laborious process. The wreckage of rejected theories is appalling; but a knowledge of what actually goes on behind what we can see or feel is surely if slowly being attained.”7

Now I think the whole drift of these statements, and particularly this last sentence, makes it abundantly plain that Dr. Hicks—and I am sure he is not alone—regards the hydro-kinetic theory of matter which he passes on to discuss, not as so much descriptive parable or ‘conceptual shorthand,’ but as veritable, conceivably perceptible, reality; in short, “what actually goes on behind what we can see or feel.” Very good. Let us now try to understand what this means.

If this primordial fluid is real, it must have some positive attributes, and it cannot be an abstraction. But it is defined as inert, incompressible, inextensible, inviscid, and structureless, all negative terms. It is useless to reply that it is quite indifferent whether we use words that are positive, or words that are negative in form; that, in fact, this primitive fluid can be equally well defined as massive, of constant density, perfectly mobile, and absolutely homogeneous and continuous. Leaving the question of mass or inertia aside for a time,—we shall have to deal with it more at length, presently,—the remaining properties are, I take it, all summed up in the one phrase ‘perfect fluid.’ And as all the fluids we know are imperfect, it might seem that the negation belongs to the known, not to the unknown. But to say nothing of the obvious impossibility of this, we find that the characteristics of an imperfect fluid, one and all, refer to experimental facts. All liquids are compressible, viscid, and more or less discrete or structural. Let me cite a witness who has some claim to speak on such a point, I mean Clifford: —”A true explanation describes the previous unknown in terms of the known; thus light is described as a vibration, and such properties of light as are also properties of vibrations are thereby explained. Now a perfect liquid is not a known thing, but a pure fiction. The imperfect liquids which approximate to it, and from which the conception is derived, consist of a vast number of small particles perpetually interfering with one another's motion.… Thus a liquid is not an ultimate conception, but is explained—it is known to be made up of molecules; and the explanation requires that it should not be frictionless. The liquid of Sir William Thomson's hypothesis is continuous, infinitely divisible, not made of molecules at all, and it is absolutely frictionless. This is as much a mere mathematical fiction as the attracting and repelling points of Boscovich.”8 Even Professor Lodge, though a sturdy upholder of the hydro-kinetic ideal, seems willing to allow the impropriety of the term ‘fluid.’ “Ether,” he says, “is often called a fluid or a liquid, and again it has been called a solid,… but none of these names are very much good; all these are molecular groupings, and therefore not like ether [the name Professor Lodge applies to this primitive medium]; let us think simply and solely of a continuous frictionless medium possessing inertia, and the vagueness of the notion will be nothing more than is proper in the present state of our knowledge.”9

Very good; again leaving aside for a moment the property of inertia, let us think simply and solely of this “continuous frictionless medium,” neither ordinary fluid nor solid. Wherein does it differ from space? Space too is incompressible, inextensible, frictionless, and structureless, and it furnishes the very form and type of a continuous medium. But whereas space is a perfect vacuum, it will be replied, our medium is a perfect plenum. But from empty space to masses in motion is a distinct step and from a uniformly filled space the step is just as distinct. So far as the realisation of any form or motion, thing or process, is her one aim, Nature ought to abhor such a plenum quite as cordially as she is said to abhor a vacuum. But the primordial medium has mass, we shall be reminded; in other words, it is inert, and inertia at least is a definite and fundamental physical fact. Let us now, then, inquire whether this remaining attribute of the universal medium renders it any more determinate, or whether, as so applied, ‘inert’ is anything better than another negation.

Inertia as a qualitative term and in its primary sense of inability or incapability is obviously negative. So Young defined inertia as the incapability of matter to alter its existing state except under the influence of some external cause. To allow that this universal plenum has inertia then does not remove its indeterminateness. Before it can be determined or differentiated in any way, some cause must intervene entirely from without, and such intervention will not admit of physical description. Such cause is of the nature of creation or miracle; it is neither a force in the sense of the attractions or repulsions by which Boscovich and Rant sought to explain matter, nor is it force in the modern sense of mass-acceleration. In other words, in the kinetic ideal of matter we shall find that the notion of mass is used with two distinct and inconsistent connotations. Abstract mechanics, as we have seen, sets out from definite masses or bodies having assignable positions, between every two of which there are dynamical transactions. Two masses, that is to say, measure each other by their mutual accelerations; in other words, mass is a strictly quantitative notion, and as such implies relation to a standard. Not only is mass in this wise always a relative quantity, but it is relative again in implicating the correlative notion of moving forces or stress between masses, which, as just said, is the only means of determining mass. If we attempt to apply the notion of mass to a universal homogeneous plenum, it lapses back into the merely qualitative notion of incapability of change evenly diffused through all immensity. And definite forces—necessarily present where there are definite masses to interact—seem here excluded. I trust I am not mistaken on this point. But it is difficult to imagine what definite forces there can be. Everything chemical or thermal or electrical is excluded, for the medium is throughout homogeneous and structureless. In like manner gravity, elasticity, and cohesion seem incompatible with absolute inviscidity and uniform density. Accordingly, to secure stability, when this medium is churned up into a labile ether it must be provided with a fixed boundary or be extended to infinity. Mathematically these alternatives may come to the same thing, though the latter, i.e. infinite extension, seems the simpler and less arbitrary of the two, again shewing how little there is to choose between a vacuum and this plenum. The properties of such a plenum, indeed, as Maxwell chanced to remark10 a year before Lord Kelvin's great hypothesis was broached, “may be dogmatically asserted but cannot be mathematically explained.” The reason for this seems simple: such a medium does not furnish even to abstract mechanics any που̑ στω̑.

However, assuming that in some ultra-physical fashion it has been whisked up into that state of turbulent motion to which Lord Kelvin has given the name of “vortex-sponge,”— this being the first step in cosmic confectionery,—let us see how this primitive mass is related to the phenomenal masses that then appear. The point I wish to urge is that neither the one nor the other conforms to the conception of mass with which abstract mechanics set out. The mass of every portion of the primitive fluid is an inalienable property of that portion. So far good, of course. Again, since the fluid is, and ever remains, of uniform density, the primitive or ‘actual mass’ of every portion is proportional to its volume. A vortex-ring is such a portion. But now its mass as measured by its mechanical effects is not simply proportional to its volume; in determining this ‘effective mass,’ the ‘strength’ of the vortex, i.e. its rotational motion, is also a distinct and independent factor. In short, this quasi-mass, or “nonmatter in motion,” depends upon a number of conditions, of which the real or primitive mass is only one. Such quasi-mass is therefore not an inalienable property in the sense in which primitive mass is such. For instance, though the volume of a vortex is constant, and therefore its primitive mass also, its configuration is liable to vary—in which fact of course lies the chief merit of the vortex-atom. But on these variations in its configuration depends the extent to which other portions of fluid are carried along with the vortex, as it moves onwards. Thus, while its primitive mass is invariable, its effective mass may vary with its motion and configuration.

We are brought, in short, to this paradoxical result: First, mechanical mass, the mass we know, is resolved into a mode of motion of some ultra-physical mass not directly capable of mechanical transactions, a mass that we therefore do not, and cannot, know as such. Given so much space, there is given also so much of this ultra-physical mass; but how much or how little nobody can say. Our scientific teachers have trespassed unawares beyond the limits of the phenomenal, and we find ourselves bowing down to a ‘fetish’ after all, none other indeed than that hoary idol of metaphysics, τò ἄπειρον, materia prima,11 qualitatively indeterminate and quantitatively indistinguishable from space. Secondly, a mechanical, effective, or apparent mass, instead of being a constant and ultimate physical quantity, as at first defined, proves, so Professor Hicks tells us, “a much more complicated matter, and requires much fuller consideration than has been given to it.” It may even, he thinks, “depend to some extent at least on temperature, however repugnant this may be to current ideas.” Thus in this endeavour to carry through the application of abstract mechanics to all physical phenomena, the conception of mass proper has got pushed over the brink of the sensible and empirically verifiable, and seems in danger of being lost in those terrible ‘metaphysical quagmires’ at which, as we have seen, the reputable physicist shudders. So now, instead of having this conception to the good in explaining or describing physical phenomena, the semblance of mass has itself to be accounted for; and this, as we have just been told, is a very complicated business “requiring much fuller consideration than has been given to it.” The impasse which thus threatens to end the kinetic ideal of matter was clearly seen by Maxwell and is admitted by Lord Kelvin. In the article ‘Atom’ in the Encyclopædia Britannica Maxwell thus criticises it: “Though the primitive fluid is the only true matter, according to the kinetic ideal that is to say, yet that which we call matter is not the primitive fluid itself, but a mode of motion of that primitive fluid. … In Thomson's theory therefore the mass of bodies requires explanation. We have to explain the inertia of what is only a mode of motion, and inertia is a property of matter, not of modes of motion.” Lord Kelvin himself, in concluding his lecture on ‘Elasticity as a Mode of Motion,’ acknowledges that “this kinetic theory of matter is a dream and can be nothing else, until it can explain,” not only the “inertia of masses (that is, crowds) of vortices,” but also gravitation, chemical affinity, and much besides. His only ground of confidence appears to be the “belief that no other theory of matter is possible.”12 But this was in 1881; and one cannot help wondering whether Lord Kelvin's confidence in his theory has increased or diminished in the meantime. Some among the younger generation of physicists prefer, as I mentioned in the last lecture, to abandon the attempt to reduce all physical phenomena to a connected mechanism based solely on the Newtonian laws. Many of them look to find a better way by taking, not mass, but energy, for the fundamental notion. Before we pass on to this, however, it will be well to try to gather up the main results of our inquiry into the mechanical theory so far.

We have distinguished three branches of science which, though distinct, are closely connected and often confused: (1) Pure, or Analytical Mechanics; (2) Mechanics applied to Molar Physics, which might be called Molar Mechanics; and (3) Mechanics applied to Molecular Physics, or Molecular Mechanics. The first is in the strictest sense an exact science based on certain fundamental assumptions and definitions. We have here rigorous calculation, but not concrete measurement: ideas, but not facts. The other two rest in part on observation and experiment, which yield approximate measurements, probable values, i.e. averages and means corrected by the help of that—for the student of knowledge—most wonderful instrument, ‘the logic of chance.’ In the exact sciences, within the limits of our powers and subject only to the laws of thought—we are complete masters of the situation. Our intellectual constructions are archetypal and not ectypal. We can here give a meaning to absolute time, absolute space, absolute motion; we can here talk reasonably of the perfectly continuous, perfectly discrete, and perfectly constant. But applied to the particulars of experience such conceptions have no warrant. The Pythagorean proposition, for example, is exact and certain, apart from all physical circumstances as a proposition in plane geometry. But, as Riemann's famous dissertation suggests, it is quite conceivable that this proposition should be falsified one way in astronomical measurements, if the distances measured were sufficiently vast; and be falsified another way—in mineralogical measurements, say—if these distances were sufficiently minute. Of course we might prefer to consider our lines as not really straight. This, however, might quite well only mean changing one contradiction for another, or prove far less simple than it would be to describe the facts in terms of some non-Euclidean space. But worse than this and far less open to dispute: the most elementary conditions of absolute exactness everywhere fail us. We have no fixed points, no fixed directions, no accurate timekeeper, not one demonstrably constant property of a physical description. Even number when applied to physical phenomena is no exception, in so far as neither identity nor simplicity nor discreteness admit of more than a relative application.

Now, as a consequence of all this, if you like—as the price of its formal exactness, abstract mechanics has to renounce those higher categories, Substantiality and Causality, which bring us into touch with concrete things. The process of eliminating these categories has been slow; for the terms ‘mass’ and ‘force’ seem almost inseparably associated with substance and power, from which notions in fact they were primarily derived. But regarding the elimination as at last complete, and accepting the purely mathematical definitions of mass and force now in vogue, the bearing of this result on molar and molecular mechanics is important. The simplest and most comprehensive description of the movements, actual or supposed, that occur in nature becomes the sole aim of these sciences, not the unveiling of the mystery of matter or the knowledge of the causes of things. The logical development of this procedure we have attempted to follow in some detail, and the outcome, as we have just seen, is that we find nothing definite except movement left. Heat is a mode of motion, elasticity is a mode of motion, light and magnetism are modes of motion. Nay, mass itself is, in the end, supposed to be but a mode of motion of a something that is neither solid nor liquid nor gas, that is neither itself a body nor an aggregate of bodies, that is not phenomenal and must not be noumenal, a veritable ἄπειρον on which we can impose our own terms. I am sure this process will remind many of you of one of Alice's Adventures in Wonderland. I trust I may be pardoned for the allusion. The Cheshire Cat, you remember, on a certain occasion, “vanished quite slowly, beginning with the end of the tail and ending with the grin, which remained some time after the rest of it had gone. ‘Well! I've often seen a cat without a grin,’ thought Alice, ‘but a grin without a cat! It's the most curious thing I ever saw in all my life.’”

In this advance towards what looks like physical nihilism, molar and molecular mechanics constitute each a distinct step. The salient feature we have noted in molar mechanics is that ‘species of abstraction’ that Thomson and Tait describe as ‘limitation of the data.’ Of such abstractions we have an instance in the treatment of the constraints and connexions that limit the free motion of a particle or of the separate portions of a machine, as mere geometrical or kinematic conditions. In actual fact constraint involves friction, strings stretch, levers bend, and so on. But all these imply intermolecular forces, the investigation of which is passed on to experimental physics. Again a change in the momentum of a body may be due to any one or more of a variety of causes—gravitation, heat, chemical action, and so on. Molar mechanics considers none of these: it is concerned only with the rate of the change itself, giving, as we must remember, the name of ‘moving force’ to this effect. The various causes, as we are allowed provisionally to call them, are, as before, passed on to, corresponding departments of experimental physics. Finally the bodies moving have manifold properties. Of these all save mass and mobility are ignored, and the rest again passed on to experimental physics.

But now assume for a moment that molecular mechanics has fully accomplished the task assigned to it, I mean this mechanical interpretation of the facts of experimental physics. None of those conditions of constraint, none of those natural forces or physical properties, which molar mechanics passed on, will then be left over; all of them will have been described in terms of mass and motion. It is thus obvious that that ‘species of abstraction’ or limitation so characteristic of the methods of molar mechanics does not pertain to molecular mechanics. On the contrary, that science, if verily complete, would—we have been told—embrace in one scheme all the vast variety of physical phenomena reduced to the simplest possible form. True, its fundamental ideas would be the same as those of pure mechanics, but then we should be assured that there were no others, whereas in molar mechanics this still remained an open question. In fact this last science would itself be absorbed; inasmuch as a body of sensible dimensions would be but an aggregate of molecules, and all those of its properties, left aside as non-mechanical in the aggregate, would be referred to mechanical processes in the parts. It is allowed, of course, that molecular mechanics is not complete; and we have seen that its procedure, when seeking to express the facts of chemistry, light, electricity, etc., in purely mechanical terms is in the main hypothetical and indirect. Molecules, Atoms, Ethers, Prima Materia—one and all are hypothetical. “Nevertheless,” say the naturalists, “they are thoroughly sound hypotheses and their scientific value is enhanced daily both by known facts that they are continually assimilating, and new facts that they are continually revealing. We realise that there is still much to do, but at the same time we are confident that ‘no other theory of matter is possible.’ Our scheme is therefore regarded as established in principle despite important gaps in detail.”

Now it is this advance—from dynamical theory, as a branch of pure mathematics, through molar mechanics, as an abstract application of that theory, on to molecular mechanics, in which all physical phenomena are subsumed under it—that vitally concerns us. A science which at the outset is simply formal and quantitative seems in the end to yield the ideal of concrete physical existence, what Kant might have called the omnitudo realitatis of the physical world; and this becomes, for those to whom the physical world is primary and fundamental, the supreme and only omnitudo realitatis that science can ever know. Here, then, we have that advancing tide of matter which, as Huxley says, “weighs like a nightmare on the best minds of these days.” But surely if our account of this transformation of pure mathematics into concrete physics is correct, the baleful spectre should be dispelled, and that without any recourse to such an agnosticism as Huxley's. The mechanical theory, in a word, as I have already hinted, refutes itself by proving too much. Or, to put it otherwise, and more fairly: the mechanical theory, as a professed explanation of the world, receives its death-blow from the progress of mechanical physics itself.

As long as the ideal of matter consisted of the “solid, massy, hard, impenetrable, movable particles of various sizes and figures” (such as Newton supposes in his Opticks), maintained in various states of vibration, rotation, and translation by their mutual encounters; so long this ideal of matter answers to Newton's conception of a vera cause. But the simple, atom or centre of force of Boscovich, and the primitive fluid of Lord Kelvin, are not verœ causœ: we must not call them fetishes, but they are assuredly fictions. To Newton's particles we might, perhaps, apply Dr. Hicks's words: “They lie outside the scope of our bodily senses;… not because they are imperceptible, but because our senses are too coarsegrained to transmit impressions of them to our minds.” To bodies wholly devoid of extension, or to a plenum wholly devoid of differences, such language cannot be applied. The process of analysis up to the stage of the chemical or physical molecule, though hypothetical and indirect, may yet be regarded as real analysis; and had the hypothesis of extended molecules proved adequate, the mechanical theory might, so far as science goes, have held its ground. Extended, solid, indestructible atoms have always been the stronghold of materialistic views of the universe. But, unhappily for such views, the hard, extended atom was not equal to the demands which increasing knowledge made upon it. Then, as we have seen, encouraged by Newton's essentially descriptive conception of distance-action, the old atom shrank up gradually, surrendering all its extension, rigidity, and elasticity, till it became identical with the entirely formal conception of analytical mechanics, that, viz., of a mass-point as a centre of force. But this later analysis, though still hypothetical, had no longer any conceivable physical counterpart. The supposition that it had was due solely to that failure to realise the purely descriptive character of mechanics which its increasing mathematical formulation and its liberation from the categories of substance and cause have now made clear. It fell to Père Boscovich decently to inter the genuinely mechanical theory as an explanation of physical phenomena. There was no rest for the old atom till it took this ghostly form of a mass-point, and thenceforward it was a dynamical fiction, pure and simple.

Lord Kelvin's brilliant hypothesis of vortex-atoms, if regarded as an endeavour to resuscitate indestructible and extended atoms as realities, and to provide a medium for their interaction, must be pronounced a failure too. Boscovich resolved the palpable atom into an idea; Lord Kelvin seems to attempt the converse and far harder feat of calling back this atom from a “vasty deep” so dangerously like pure being as to be, phenomenally, pure nothing. The endeavour to attribute mass to this continuum is as if one should let one's plummet drop in the hope of sounding a fathomless sea; we lose a simple conception, and have a complex one left on our hands instead. But now comes Dr. Hicks to persuade us that we gain more than we lose: “If it should be found that all phenomena are manifestations of motion of one single continuous medium, the idea of force will be banished [the relative idea, that is, of which mass is the correlative]… and the study of dynamics will be replaced by the equation of continuity;” for “where all the matter is of the same density the motions are kinematically deducible from the configuration at the instant, and are independent of the density.”

These remarks are most opportune. If we consider them for a moment, they ought to satisfy us that we are not penetrating beyond what we see and feel to anything that actually goes on behind the too coarse grained veil of sense. They serve to shew, on the contrary, that the kinetic ideal also is but a fiction of the mathematician, a descriptive symbol, and not conceivably a presentable fact. Now there is a certain philosophical doctrine, both psychologically and epistemologically of fundamental importance, that ought to be well known in Aberdeen13—I mean the doctrine of the relativity of knowledge. The range of this doctrine may be very much a question, but at least no one will deny that it applies here. See then to what it leads. Everything perceptually real, everything phenomenal, whatever can be an object of possible experience, implies difference and change. But we have left all sensible qualities except density behind us; and this, though retained, is to admit of neither difference nor change. “Idem semper sentire et non sentire ad idem recidunt,” says the doctrine of relativity. For any conceivable experience then this density is as nothing. Moreover, according to the kinetic theory, the motions are independent of it. Why then is it retained? Apparently to stand between us and nonentity. It secures for us that “idea of stuff or substance which,” Professor Tait tells us, “the mind seems to require”—well for comfort!14 It is then das reine Sein of our present universe of discourse. Or it is the ‘Achilles heel’ of reality, left when all the rest of the physical world has been dipped in the Styx. “But why,” asked an intelligent child, “did not Thetis dip Achilles twice?” Now Dr. Hicks appears to have had that much foresight in agreeing to let go dynamics and to abide by the equation of continuity. For dynamics and mass must surely vanish together, and we have properly only kinematics left. Nevertheless there remains one stipulation that kinematics does not warrant—there must be no discontinuity in the motions on two sides of a geometrical boundary. The vortices, in other words, must not spin and leave the medium unaffected; and so the medium, being involved in the movement of one vortex, must in turn affect the movements of another. And thus with this proviso the whole becomes, as we may say, one vast quasidynamical or rather quasi-kinematical system. For it is allowed,15 I believe, that the existence of surfaces of finite slip is not precluded by the bare conception of a uniform frictionless medium. Imagine such an ideal fluid if you can, and the question whether a vortex in it will or will not affect the fluid outside the vortex is altogether indeterminate. It may do either or neither, sometimes the one and sometimes the other. Why then is this condition of motional continuity imposed from without? Simply to make the thing work mathematically, that is to say, to insure connexion and continuity between one kinematical configuration and another. Without it we might have vortex-atoms as before, but not “actions excited by these vortices on one another through the inertia of the fluid which is their basis.”16 Such mutual regard is not then a direct consequence of the common plenum. In fine, then, this additional property of motional continuity is asserted, though it cannot be deduced, in order to make possible a kinematical scheme that replaces, as Dr. Hicks says, the dynamical laws that can then be left behind.17

It may be that this exposition by the President of the Physics Section of the British Association sounded rash, or at least premature, to the distinguished physicists who heard it. But it must certainly be impressive to any humble outsider with a philosophical bent. It exhibits strikingly the complete logical outcome of the problem of mathematical physics, as formulated by the Kirchhoff school; and all the more strikingly because this consequence is here worked out, as it were unconsciously, by one who, unlike Kirchhoff, seems to suppose that he is all the while getting nearer to “what actually goes on” in the real world. The tendency to extend kinematics at the expense of dynamics seems inherent in this new conception of physics. But the sounder the conception, the more this tendency may be expected to assert itself spite of contrary prepossessions, and the more effectively will such prepossessions be dispossessed.

Now it is entirely upon these uncritical prejudices, as we may fairly call them, that the mechanical theory of the world rests. The more they are discredited the more it is discredited through them, and this, I believe, the history of science will amply show. The transference of motion by impact, for example, as when two billiard balls collide, seems the type of plainness, and so long as this and other equally familiar experiences were accessible to the imagination, it seemed still to retain its grasp of the real spite of ‘the cloud of analytical symbols.’ The triumph of the undulatory, over the corpuscular, theory of light, was a blow to such realism; for an imponderable ether was not easy to conjure up by imagination. Still, after all, waves are familiar and it was only the ‘undulating agency’18that was obscure. But a severer blow overtakes us in what we might call the demolition of the chemical atom as an assured stronghold of the realistic imagination. And when both chemical elements and luminiferous ether are resolved into motions of a medium, ‘the dynamics of which is not the dynamics of ordinary matter,’19 realism seems fairly routed. But stranger still, imagination has become itself a traitor to mechanical realism—I refer, of course, to such ingenious mechanical analogies as those, for example, by which Maxwell succeeded in elucidating electromagnetism. Analogy is an important aid to description, though powerless to prove existence. Nevertheless, as I had occasion to remark in the last lecture, even the ablest men are apt to see more in analogy than this; and it speaks volumes for Maxwell's strength of intellect that, acute as he was in the discernment of helpful analogies, he seems never to have been led away by them. But it is a case in which there is safety in numbers. A thinker familiar with many analogies is less likely to be betrayed by them than a thinker whose mind is enchanted by one. Now Boltzmann, in an instructive paper on the Methods of Theoretical Physics from which I have already quoted once or twice, gives many instances of surprising and far-reaching analogies that have been discovered within the last half-century between physical phenomena apparently quite unlike; as if nature had “built up the most diversified things after exactly the same pattern.” “As the analyst dryly observes, the same differential equations hold for the most diversified phenomena.” And no great wonder if the analyst previously made up his mind to see the most diversified phenomena merely as cases of motion, to be described in the simplest and most comprehensive manner. The logical goal of such a project, I conclude then, is—if I may so say—to minimise the inevitable ‘matter’ of phenomena and to bring all the diversity possible under the ‘form’ of motion. This goal is already set before us in the kinetic ideal of matter, where dynamics is all but sublimated into kinematics. So much so indeed, I may remark by the way, that even the motion is absolute, and not merely relative motion; for every motion is strictly a motion of the medium, and this is infinite and all there is. Now, as soon as we are asked to entertain the notion of absolute motion, we may satisfy ourselves that we have left everything phenomenal behind us and are once again entirely in the region of the abstract conceptions of exact mathematics. And the medium itself, though infinite and all there is —nay, because of this, for it does not allow even the distinction of body and space—is indistinguishable from nothing. The whole ideal, it seems to me, if it be meant to set before us what verily is and happens, was refuted long ago by Leibnitz in the following sentences of the Monadology (§ 8): “If simple substances did not differ at all in their qualities, there would be no way of perceiving any change in things, since what is in the compound can only come from the simple ingredients, and if the monads were without qualities they could not be distinguished the one from the other, since also they do not differ in quantity. Consequently, a plenum being supposed, each place in any movement could receive only the equivalent of what it had before, and one state of things would not be distinguishable from another.”

We smile at the critical simplicity while admiring the boldness of the Pythagoreans, according to whom, as Aristotle tells us, “Number is the essence of all things; and the organization of the universe, in its various determinations, is a harmonious system of numbers and their relations.” Enough perhaps is known of the Pythagoreans and their tenets to shew that they had no pure science of number, but that such arithmetical knowledge as they had was encumbered by concrete and fanciful associations with numbered things. May we not apply the moral to the mechanical theory of the universe, and say that the more clearly the purely mathematical character of mechanics is realized, the more absurdly inadequate that theory becomes? A science that can only offer us as its ultimate scheme of the universe the inconceivable ideal of continuous motion in an unvarying plenum, is surely as incompetent as arithmetic or geometry to furnish a concrete presentment of a real and living world. Its essentially formal character has become increasingly evident with every improvement in its methods. Galileo and Newton made many experiments, and their works abound in diagrams; but I am not aware that either Lagrange or Laplace20 ever tried an experiment, while Lagrange is said to have boasted that his Mécanique analytique did not contain a single figure. This science, then, which has gradually rid itself of the categories of substance and cause, which works entirely with abstract quantities, expressing its conditions in equations and its results in equations, does not, and cannot, yield any direct knowledge concerning real things. When employed to describe them, its application is restricted absolutely to the one quantitative aspect with which it deals,—the motions of mass-systems. It has no scientific status except where such motions are either (1) given, or (2) inferred, or (3) assumed. In the first case its results, though necessary and exact in themselves, become at once hypothetical and approximate in their application; the ideal simplicity and abstract isolation of theory being never found in reality. In the second case the results are more hypothetical and approximate still; for neither the particles nor the motions themselves can be directly measured. This is the region of statistical probabilities. In the third, the masses and motions are entirely hypothetical; it is no longer, strictly speaking, a case of applying pure mechanics to describe real motions. This is the region of mechanical analogies, of prime atoms and ethers, vortices and primordial fluids; the region in which, as Dr. Hicks has told us, “the wreckage of rejected theories is appalling.”

The mechanical theory of the universe, then, begins with abstractions, and in the end has only abstractions left; it begins with phenomenal movement and ends by resolving all phenomena into motion. It begins with real bodies in empty space, and ends with ideal motions in an imperceptible plenum. It begins with the dynamics of ordinary masses, and ends with a medium that needs no dynamics or has dynamics of its own. But between beginning and end, there are stages innumerable; in other words, the end is an unattainable ideal. First, we have sensible mechanisms; to these theoretical formulæ only apply approximately, their abstract simplifications being inadequate to cope with the ‘practically infinite’ complexity of the reality. A closer approximation is secured, but at the cost of new residual discrepancies, by resolving the parts of sensible mechanisms into smaller mechanisms, and the parts of these into others yet smaller in turn. Again, further approximations are made by attributing other elements of the real complexity to imaginary mechanisms of many orders. But the complexity being, as said, ‘practically infinite,’ this procedure has no prospect of ending. Dr. Hicks, for example, even when he has got as far as the chemical atom,—and that, we must remember, is a very long way,—cheerfully tells us, “The atom is much larger than a cell, and contains, practically, an infinite number of them; “a cell, I must tell you, being an imaginary box that Dr. Hicks has devised, in which a vortex of the primary medium is magically penned up to wriggle. Yet, spite of these complex mechanical fictions, no advance is yet reported towards a kinetic theory of gravitation, and very little has been done with the terrible complications of chemical affinity. The story of the progress so far is, then, briefly this: Divergence between theory and fact one part of the way, the wreckage of abandoned fictions for the rest, with an unattainable goal of phenomenal nihilism and ultra-physical mechanism beyond.21 Nevertheless, there are many who hold that the world must be such a mechanism, because they imagine themselves unable to conceive it otherwise. Such, as I understand it, is Lord Kelvin's position, for example. Others see in the situation a parallel to that of the Ptolemaic astronomy, which could not cope with increasing knowledge even with the help of new eccentrics and epicycles, freely assumed as the occasion arose. A new and simpler science of energetics is with some of these reactionaries the counterpart of the Copernican astronomy, and is to release physics from the complications in which mechanics has involved it. These are points that must occupy us in the next lecture.

  • 1.

    Kroman, Unsere Naturerkenntniss, p. 315.

  • 2.

    Die Grundlagen der Erkenntniss in den exacten Wissenschaften, p. 37

  • 3.

    Scientific Papers, vol. ii, p. 316.

  • 4.

    Properties of Matter, art. x.

  • 5.

    Scientific Papers, vol. ii, p. 321.

  • 6.

    Maxwell, Scientific Papers, vol. ii, p. 471

  • 7.

    Nature, vol. iii, p. 472; italics mine.

  • 8.

    Lectures and Essays, vol. i, p. 238 f.

  • 9.

    The Ether and its Functions, Nature, vol. xxvii, p. 305.

  • 10.

    Scientific Papers, vol. ii, p. 26 (on Dynamic Theory of Gases).

  • 11.

    Cf. Descartes, Les Principes de la Philosophie, bk. ii, art. 5.

  • 12.

    Popular Lectures, vol. i, p. 145.

  • 13.

    Being so strenuously maintained by Dr. Bain.

  • 14.

    Unseen Universe, p. 105.

  • 15.

    See letter on Vortex-atoms, by Professor G. H. Darwin, Nature, vol. xxii, p. 95.

  • 16.

    Dr. Larmor, Proc. R. S., 1893, P. 439.

  • 17.

    “It will be seen that the work is almost entirely kinematical; we start with the fact that the vortex-ring always consists of the same particles of fluid (the proof of which, however, requires dynamical considerations), and we find that the rest of the work is kinematical. This is further evidence that the vortex theory of matter is of a much more fundamental character than the ordinary solid particle theory, since the material action of two vortex-rings can be found by kinematical principles, whilst the ‘clash of atoms’ in the ordinary theory introduces us to forces which themselves demand a theory to explain them.” Professor J. J. Thomson, A Treatise on the Nature of Vortex-Rings, 1883, p. 2.

  • 18.

    Lord Salisbury, Presidential Address, British Associatiion, 1894.

  • 19.

    Larmor, Nature, vol. liii, p. 4.

  • 20.

    Note iv.—This statement, Professor Poynting tells me, must be modified in so far as Laplace was associated with that masterly experimenter, Lavoisier, in investigating specific heat and the dilatation of solids with rise of temperature. But the following sentence confirms the estimate given of him above:—“It was perhaps as much because it threatened an inroad on a cherished generalisation as because it seemed to him little capable of mathematical treatment that the undulatory theory of light was distasteful to him” (Encyclopædia Britannica, article Laplace, p. 303).

  • 21.

    This passage is quoted by Sir Arthur Rücker in his Presidential Address to the British Association in 1901, in which he seeks to defend the reality of ‘ultra-physical entities.’ I have tried to deal with his position in a Supplementary Note.