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Part 1. The Mechanical Theory

Lecture 3: Relation of Abstract Dynamics to Actual Phenomena

The characteristics of Abstract Dynamics recapitulated.

The question raised: How far, and in what sense, this science can be applied to actual phenomena. This problem illustrated from Newton's treatment of Space, Time, Motion, as (1) absolute; (2) relative.

Bearing of this distinction on the attempt to determine an actual case of the first law of motion. Various proposals considered. The question of absolute rotation especially instructive. Mach's criticisms reveal the indefinite complexity of ‘real cases.’

The mechanical theory is thus divided against itself: it cannot be at once rigorously exact and adequately real. The Kirchhoff School abandon the attempt “to penetrate to the mechanism of nature,” and see in mechanics only an instrument for ‘approximate description.’ Unconditional mechanical statements concerning the real world appear so far unwarrantable.

One of these specially discussed: the Conservation of Mass. Mr. Herbert Spencer's ‘short and easy method’ found wide of the mark. This doctrine, like other mechanical doctrines, justified mainly by its simplicity.

WE resume to-day the attempt to estimate the validity and the scope of the mechanical theory of the universe. To understand this we have had first of all to inquire into the precise import of the science of abstract mechanics or dynamics, on which that theory is avowedly founded. We have accepted the declaration of mathematical physicists in the present day that it is not the province of mechanical theory to explain phenomena by means of natural forces, but only to describe completely in the simplest possible manner, such motions as occur in nature.1 We appreciate most readily the distinctive character of pure mechanics, as thus defined, if we approach it from the side of kinematics. Kinematics is held to suffice for the description of any actual or possible motion of bodies, regarded as moving figures of constant or varying shape. If there are some motions too complex for kinematic treatment in the present state of that science, the defect is one that mechanics can do nothing to remove. But “the motions that occur in nature” are frequently, and, it is supposed, are always, mutually dependent. As to the character of this dependence, the most various hypotheses might be—indeed have been—formed; and when such hypotheses are sufficiently definite, as regards their space and time elements, their kinematical consequences can be deduced. The kinematical problems thereby entailed might be appalling in comparison with those required by the simple assumptions to which, after many trials, Galileo, Huygens, and Newton, the founders of modern dynamics, were led. By means of the conception of mass the notion of quantity of motion, or momentum, was made definite by Newton, and the so-called laws or axioms concerning momentum formulated. According to these the rate at which their momentum changes, when two masses are in the state of mutual stress, is always equal in amount, their motions taking place in opposite directions along the line joining them, the result being that the momentum of their common centre of mass remains unchanged.

Nothing could be more sublimely simple, especially when it is remembered that these axioms involve the so-called parallelogram of forces; imply, that is, that the mutual accelerations of any two masses are uninfluenced by the presence of a third mass. Such is abstract dynamics; and, regarded from within, its exactness is as impressive as its simplicity. Not only is it clear of such ‘bottomless quagmires’ as substantiality and causality, conceptions which no science has ever yet adjusted to facts; but as ‘rational mechanics’2 it is clear, too, of all induction and all experiment, resting wholly, as truly as any formal science does, on its own fundamental definitions and axioms. The only space or time or motion that it knows is what Newton called absolute, true, and mathematical, and sharply distinguished from the relative spaces, times, and movements of our perceptual experience.

How far, and in what sense, this pure mechanical science can be applied in the phenomenal world is now for us the vital question. Unhappily the authorised teachers of physics seem only recently to have waked up to the possibility of such a question at all. The only ‘applied mechanics’ they seem aware of is that of the mechanician and the engineer. While admitting readily that the astronomer applies geometry and trigonometry in his investigations, they talk as if he were entirely in the region of pure theory as soon as he proceeds to discuss celestial movements. Newton at all events knew better than this, even if he realised the difficulty of the transition less than many now do. Let me quote a few sentences from the Principia in illustration.3 First, as to time: “Absolute, true, and mathematical time, in itself, and from its own nature, flows equally, without relation to anything external; and by another name is called Duration.… The natural days are truly unequal, though they are commonly considered as equal and used for a measure of time. Astronomers correct this inequality that they may measure the celestial motions by a more accurate time. It may be that there is no equable motion, whereby time may be accurately measured. All motions may be accelerated and retarded; but the flowing of absolute time is liable to no change. Duration…remains the same, whether motions are swift or slow or none at all: therefore this duration is properly distinguished from its sensible measures; and from them it is collected by means of an astronomical equation.”

Again, as to space: “Absolute space, in its own nature, without relation to anything external, remains always similar and immovable.” “For the primary places of things to be moved is absurd. These are therefore absolute places; and translations only out of these are absolute motions. But, because the parts of space cannot be seen, or distinguished from one another by our senses, therefore in their stead we use sensible measures…and that without any inconvenience in common affairs: but in philosophical disquisitions, we must abstract from the senses. For it may be that no body is really at rest, to which the places and motions of others may be referred.… It is possible that in the regions of the fixed stars or far beyond them, there may be some body absolutely at rest; but yet [it is] impossible to know from the position of bodies with respect to one another in our regions, whether any of them do keep the same position to that remote body or no. It follows [therefore] that absolute rest cannot be determined from the position of bodies with respect to each other in our regions.”

Lastly, as to motion: “Absolute motion is the translation of a body from absolute place to absolute place; and relative motion is the translation from relative place to relative place.” “If a place is moved, whatever is placed therein is moved along with it.… Therefore all motions which are made from places in motion, are only parts of entire and absolute motions: and every entire motion is composed of the motion of the body out of its first place, and of the motion of this place out of its place, and so on, until we come to some immovable place, as in the example of the sailor before mentioned [who was supposed to move relatively to his ship which moved relatively to the earth, which in turn, moved relatively to the sun, and so on and on]. Wherefore entire and absolute motions can be no otherwise determined than by immovable places.… It is indeed a matter of great difficulty to discover and effectually to distinguish the true motions of particular bodies from the apparent: because the parts of that immovable space, in which motions are truly performed, do not come under the observation of our senses. Yet the case is not altogether desperate; for arguments may be brought, partly from the apparent motions, which are the differences of the true motions; partly from the forces, which are the causes and effects of true motions.”

One can readily gather from statements like these that Newton saw no difficulty in working out problems in which the time should flow at a constant rate, and in which motion from absolute place to absolute place was at once and effectually determined. The position of mechanical theory is in this respect precisely on a par with that of geometry. The description of the circle, say, is easy and exact, but accurately to describe the figure of any real object is an impossibility. So it is with the fundamental quantities concerned in physics.

It is impossible to find in nature or artificially to construct an accurate timekeeper. The physicist simply has to collect the true time from its ‘sensible measures,’ to use Newton's phrase, as nearly as he can. Experience provides us with innumerable instances in which processes seemingly identical in character and severally independent, are again and again repeated in such wise that the number of repetitions of one kind of process is found to bear an approximately constant ratio to the number of repetitions of another and contemporaneous series. The solar day, the lunar month, the solar year, so far as we may regard them as independent events, are instances of such isochronous series of the natural sort; the periods of waves of light or of waves of sound are other instances; while the vibrations of a given spring or a given pendulum are cases of artificial isochronous events. The comparison of a number of such series—aided by dynamical reasoning, whereby certain disturbances can be ascertained and corrected, and aided again by the theory of probability in eliminating errors of observation—results not in the attainment of a measure flowing equably without regard to anything external, but in the best mean value possible in our restricted circumstances. Between such mean time and absolute time there is a difference, that is certain; and that difference is, for the mechanical theory, of the nature of error or defect. It is immaterial to the question we have in hand whether absolute time is also real or is ideal only. It is at least ideal, and the fact that the physicist has to leave this ideal behind him when he proceeds to apply abstract dynamics to natural phenomena is the fact to be noted.

Turning to space, the same fact meets us again. Instead of the immovable space, the fixed axes, the primary places of mathematical theory, we have that indefinite regress from relative place to relative place, which renders the attempt to ascertain the so-called true motions of particular bodies, as Newton allows, “well-nigh desperate.” Consider, for example, a case falling under the first law of motion. According to this law the motion of a body free from external forces is uniform in magnitude and direction. The mathematician has no trouble with this. He can always specify the axes to which he refers, and plot out diagrams of velocity in his paper space. But when we pass to empirically given space, where is the place to which the direction of a body moving under the action of no forces is referred? “A number of writers,” says Professor MacGregor in a recent article, “have attacked this problem, and left it only half solved.”4 Newton's forlorn suggestion that possibly in the region of the fixed stars, or far beyond, there may be a body absolutely at rest, to which the positions and motions of others may be referred, has been revived. In favour of assuming this fictitious Body Alpha, as it has been called, it is urged that such a body provides an escape, in thought at all events, from the hopeless confusion of relative motions to which there is no end.5 But ideally this Body Alpha is not wanted, and practically it is useless. Another and less chimerical method that has found more favour begins, not by asking for a body absolutely at rest as a fundamental point of orientation, but by asking for an “inertial system.” To constitute such a system it suffices to have three particles projected at the same instant from one position, and each left free to move, uninfluenced by force. Then, provided they do not all move in one straight line, it is geometrically possible to find axes, referred to which they will all three move in straight lines. Referred to such a system, the path of any fourth body moving free from force will be a straight line.6 But this again is obviously theoretical, and so far superfluous. Practically it is as impossible to ascertain that a body is absolutely free from forces as it is to ascertain its direction relatively to the Body Alpha, the presumption being indeed that no such body, unless it be the universe as a whole, exists. Yet a third method has been proposed of answering the question: Relatively to what, is a body free from constraint moving uniformly in a straight line? The answer according to this method, which has been adopted by Professor Tait, is, “Relatively to any set of lines drawn in a rigid body of finite dimensions, which is not acted on by force, and which has no rotation.”7 Here again it may be objected that it is impossible to find such a body, for if the universe is a single mechanical system, there is no such body to find.

But none the less this method brings to our notice a topic keenly canvassed nowadays among physicists, which is of extreme interest; so that I trust I may be pardoned for meddling with it. Newton believed that he had shewn, first by experiment, and then by theoretical reasoning, that “there is,” as he puts it, “only one real circular motion of any revolving body…whereas relative motions in one and the same body are innumerable.” Thus, if two bodies in an immeasurable void were found to approach, there would be no means of determining which was moving. But if the two bodies were connected by a cord, it would be possible, though their distance remained unchanged, to determine whether they were revolving or not. To settle this question it would be sufficient to ascertain the presence or absence of tension in the cord. Accordingly it is argued, as by Professor Tait, that a body not rotating will provide us with fixed directions in space, constitute a sort of absolute compass, so to say; and by the help of Newton's physical test it can be ascertained whether a body has rotation or not. Here, then, we seem to have something absolute, an exception to the supposed invariable relativity of everything phenomenal. But so far we have been given only a purely hypothetical case—a single system in an immense void. Newton's actual experiment consisted in rotating a bucket of water by strongly twisting a cord suspending it, so as to make the bucket spin rapidly. At first, when the bucket alone rotates, the surface of the water remains flat, although relatively to the bucket it is not at rest; whereas, by the time the water revolves along with the bucket its surface has become concave, thereby evidencing “real circular motion,” to use Newton's phrase, notwithstanding that the bucket and the water by this time are at rest relatively to each other. Finally, when the bucket has ceased to revolve, the surface of the water continues concave some while longer, because “its endeavour to recede” from the axis has not yet ceased. “Therefore,” says Newton, “this endeavour does not depend upon the translation of the water in respect of the ambient bodies, nor can true circular motion be described by such translation.” In other words, as Kant remarks, “a motion which is a change of external relation in space can be given empirically, although this space itself, is not empirically given, and is no object of experience—a paradox deserving to be solved.” Kant's own solution is of interest in its way, but it does not help us much, for it leaves the paradox in the main as he found it. But I will ask your attention instead to the much more trenchant criticism of Mach, as this will serve to illustrate the epistemological difference between abstract science and its empirical application, which is our immediate theme.

First of all let us note the difference between Newton's theoretical instance and his experimental one. In the purely hypothetical case we imagine a single mass system in an immense void, and it is shewn under what circumstances, provided the Newtonian laws of motion are assumed, the rotation of such a system could be demonstrated. In the real case, which is meant to verify this deduction, we are confined entirely to experimental methods. But now in this case, over and above the rotating mass of water, we have not only the mass of the bucket, but we have also the masses of the earth, of the rest of the solar system, and of the so-called fixed stars. Now, says Professor Mach, “Newton's experiment…only shews us that the rotation of the water relative to the sides of the bucket occasions no perceptible centrifugal forces, but that such forces are occasioned, when the water rotates relatively to the masses of the earth and the other heavenly bodies.”8 Experimental canons then at once suggest two further inquiries: Might not the rotation relative to the bucket have some effect if the sides of the bucket were enormously increased in thickness? Or again—allowing for the moment that the proposition is not absurd, at least not kinematically absurd—supposing the bucket to be fixed and the whole choir of heaven to circle round it, would there then be no sign of rotation in the water? Such experiments being impracticable—for, as Mach well says, “the universe is not presented to us twice, first with the earth at rest and then with the earth rotating”—we are left to content ourselves, as best we can, with this result: that a body with so-called absolute rotation is a body rotating relatively to the fixed stars; and that a body without rotation means a body directionally at rest, not absolutely, but relatively to the fixed stars.

Returning now for a moment to Newton's hypothetical case, it is obvious that a physicist actually confined to such a system, before he could begin experimentally to apply or to verify the Newtonian laws of motion, would find himself face to face with the very difficulties we have considered. Positions and directions must be independently determined before dynamical investigations are begun. To assume the laws of motion in order to fix directions and then to use these directions in order to establish the laws would be obviously fallacious. From such a logical circle abstract dynamics is free, because the physicist has here the complete command of ideal space, as is shewn by his diagrams on paper; and because he has not to prove the laws of motion, but merely to deduce their theoretical consequences. Newton's absolute rotation is then, like his absolute time and absolute space, not real but ideal, not sensibly or empirically given but intellectually conceived or constructed, not ectypal but archetypal, as Locke says of all purely mathematical ideas.

This becomes clearer, if we consider the difference between the two cases from another side. The hypothetical case is that of a finite system in an immense void; all the rest of the universe is supposed to be eliminated. In the real world we may ignore, but we cannot exclude. Thus, as already said, it is allowed that—except by accident—there is probably no body in the state described in the first law of motion, in fact, if the master generalisation of physics, the law of universal gravitation, is to be accepted, how can any particle of matter “be left to itself”? By a free particle, or a particle left to itself can only be meant a particle at an infinite distance from any other particle, and in this sense accordingly writers on abstract dynamics sometimes define the phrase. But if we could come across such a particle in actual experience, it is obvious that nothing could be said about it; spatial perception of any kind would necessarily be absent in such circumstances. In dealing with the actual world, however, the facts that meet us first are those to which Newton's second and third laws apply, and the law of inertia becomes but a special case of these. Setting out from these laws, then, instead of attempting to affirm anything concerning the movement of a particle alone in absolute space, it seems to me as a mere question of scientific taste and logic better to proceed in Mach's fashion. Instead of saying that a particle moves without acceleration in space, Mach would say that the mean acceleration of such particle relatively to the other particles in the universe, or in a sufficient portion of the universe, is zero.9

As it is obviously impossible to complete the summation required to ascertain this mean exactly, such a statement has the advantage of keeping prominent the approximate character of references to the directions of certain stars as fixed directions. The reference to fixed terrestrial objects, which sufficed for such observations as led Galileo at first to formulate the law of inertia, is now replaced by this reference to fixed stars; but even this direction is known to change in the course of ages. Another advantage of Mach's more concrete statement, then, is that it impresses us, as he remarks, with the very complicated character of just those mechanical laws that appear the simplest. Suggested by incomplete experiences in the first instance, they lose the exactness of mathematical theory when we proceed to apply them to experience again. The manifold particulars left out of account in our abstract simplification are still there on our return to confront us anew. The insight that a pure theory has given may enable us to deal with them more effectually; it cannot justify us in ignoring their existence. Now by good fortune, not from any necessity in the constitution of things, it is found that within certain limits of exactness many of these particulars of experience are so similar, that to deal with any one appears to suffice. One result of this apparent multiplicity of identicals is that, seeming to be independent of any one, we presently suppose ourselves independent of all; when to be absolutely exact we are independent of none. In applying the law of inertia to terrestrial bodies, for example, there are innumerable landmarks from which to estimate direction; if one or more become unsteady or disappear, there are still plenty of others left. So with celestial objects; if one fixed star should some day “pale its feeble light” or be found careering across the sky, there are still multitudes remaining to keep their accustomed stations. Now, it is our familiarity from time immemorial with this plenitude of possibilities that leads us to convert these several singular contingencies into a collective contingency. We then assume that, as we are independent of any one empirically marked position in space, we are independent of all. In other words, the absolute space of abstract conception is supposed to underlie the empirical space that we perceive. But now imagine, as Mach suggests, that the earth were the scene of incessant earthquakes or that the stars behaved like a swarm of flies: how should we apply the law of inertia then? Well, but to those who mean seriously to handle the universe as a mere problem in abstract dynamics we must reply that the earth is the scene of incessant convulsions and the fixed stars are like a swarm of flies. The costliness of the devices to eliminate terrestrial oscillations in certain attempts at experimental precision and the elaborate calculations to unravel the ‘proper motions’ of the less distant stars are plain evidence of the truth of this seemingly extravagant statement.

It would seem then that all bodies may be really implicated in every case of movement observing the law of inertia; not one only, as the abstract theory assumes. What a single body would be or do if it were not for other bodies, no one can say. Unless indeed they are prepared with Stallo to say boldly, it would be nothing and therefore could do nothing. “A body,” he says, “cannot survive the system of relations in which alone it has its being; its presence or position in space is no more possible without reference to other bodies than its change of position or presence is possible without such reference.… All properties of a body which constitute the elements of its distinguishable presence in space are in their nature relations and imply terms beyond the body itself.”10 In abstract theory, then, we may introduce first one particle and then another, each moving in given directions in absolute space; and we may talk of their speed as measured by absolute time flowing equably without relation to anything else. But, in reality, nothing of this kind is accessible to us.

It is easy to see that the mechanical theory is here divided against itself, and in this state cannot stand. Experience compels it to admit the thorough-going interdependence of all bodies, while mathematics tempts it to suppose that it is possible to deal with bodies independently and apart. The bodies which mathematics would regard as isolated wholes are but undetermined fragments of what is really indivisible, abstract aspects that never exist alone. On the one side is the ideal simplicity and completeness of a mathematical creation; on the other an illimitable complexity of relations without beginning, without middle, and without end. Now I presume nobody will blame the physicist for insisting on the relativity of all motion, the relativity of all time-measures, which practically depend on motion, or the relativity of all determinations of mass or inertia. But we have a right to demand logical consistency: if he abjure absolute terms he must abjure absolute statements. He must not confound his descriptive apparatus with the actual phenomena it is devised to describe. The apparatus consists, in general, as we have seen, of absolute time, that is, an independent variable flowing at a constant rate; of absolute motions, that is, motions referred to axes completely defined and thought of as fixed; of bodies that by definition are masses and only masses, absolutely determinate and unchangeable, and constituting together a mechanical system that is independent and complete. Of this general form of apparatus there may be several varieties, but that will be accounted the best which affords the simplest and completest description of actual movements. We cannot be sure that there is any a priori necessity about the particular mechanical principles of Galileo and Newton; from other fundamental definitions consequences equally exact might be deduced. As this is an assertion that to many may seem unwarranted, let me hasten to say that I do not make it without good authority; I will quote one such out of many. In an essay on the Methods of Theoretical Physics, Boltzmann, referring with approval to the changes introduced by Kirchhoff, thus proceeds: “Whether, with Kepler, the form of the orbit of a planet and the velocity at each point is defined, or with Newton, the force at each point, both are really only different methods of describing the facts; and Newton's merit is only the discovery that the description of the motion of the celestial bodies is especially simple if the second differential of their coördinates in respect of time is given.”11 In either case, and in every case, then, we have only mathematical description. “The whole difficulty of philosophy,” said Newton, in the Preface to his Principia, “seems to consist in investigating the powers of Nature by means of the phenomena of motion.” Many of his successors have abandoned the enterprise. To quote Boltzmann again: “The view [has] gained ground that it cannot be the object of theory [i.e. of science] to penetrate the mechanism of Nature, but that, merely starting from the simplest assumptions (that certain magnitudes are linear or other elementary functions), to establish equations as elementary as possible which enable the natural phenomena to be calculated with the closest approximation.” Equations, not explanations, approximation, not finality, and the simplest method the best: in such wise has the modern science of dynamics narrowed its scope. And the criterion of simplicity, it must be remembered, is in the main subjective, not objective. Our limited capacities make economy a consideration. But for such limitation, indeed, it is difficult to see why we should cumber ourselves with a descriptive apparatus of any sort. It is surely then a thoughtless prejudice to forget that the capacity to calculate and compute—though, as Laplace boasts, it renders the human species superior to the animals, and is the foundation of our glory—is also still, like apparatus generally, essentially a mark of limited powers. Regarded in this light it becomes very much a question whether the Newtonian scheme is even the simplest; indeed, other schemes, professedly simpler—and what, if true, is of greater moment, more comprehensive—are already in the air. If human capacities are limited, they are not stationary. As Kirchhoff remarks: “A description of certain phenomena, though it be indubitably the simplest we can now give, may in the further progress of science be superseded by another simpler still. Of such like changes the past history of mechanics furnishes instances in plenty.”12 Still this question of comparative simplicity does not concern us save as it may serve to impress two points. First, the difference between the means of description, “the conceptual shorthand,” as Professor Karl Pearson happily styles it, and the perceptual realities it is devised to symbolise and summarise. Secondly, the absence of finality. A possible form of description is not enough, it must be shewn to be the only one possible, the only one that the phenomena themselves allow, before it can be held to have passed out of the region of hypothesis into that of objective truth.13

The conclusion then to which we are led is plain. The application of abstract mechanics to real bodies is throughout hypothetical, and absolute or unconditional mechanical statements concerning the real world are therefore unwarrantable. There are no processes in the real world that are certainly entirely mechanical, mechanical in the sense, I mean, of those movements of sensible masses from which Galileo and Newton inductively inferred their well-known laws. The thermal, chemical, electrical, magnetic, and other processes that as a rule not only accompany but modify such mechanical movements may admit of complete and simple description in purely mechanical terms. But there is no necessity that they should. Newton saw reason to hope for it, however. In the Preface to his Principia, he justifies its title as Mathematical Principles of Natural Philosophy by referring to the motions of the planets, the comets, the moon, and the sea as deduced from gravitational forces by propositions that are mathematical. He then adds, “I wish we could derive the other phenomena of nature from mechanical principles by the same kind of argument.… But I hope that the principles here established will afford some light either to this, or some more perfect method of philosophy.” It is to this subject that we must pass in the next two lectures, and we shall then have an opportunity of inquiring which of Newton's alternative hopes is the more nearly realised: the resolution of natural phenomena that are not obviously mechanical into mechanisms, or the advent of some more perfect method embracing both. But either way our main conclusion will, I believe, still remain good.

There is one absolute statement frequently advanced by modern physicists that flagrantly transgresses the limits of a purely descriptive science, the statement, I mean, that the mass of the universe is a definite and unchangeable quantity. Such partial and approximate evidence as experience affords in favour of such a doctrine seems to be derived ultimately from the facts of gravitation. Astronomical observations of planetary motions and chemical measurements with the balance justify the working hypothesis that such sensible masses as we know are constant within the limits of our experience and unalterable by any means in our power. Thus has been suggested the addition to abstract dynamics of a principle not explicitly formulated by Galileo or Newton, that, namely, of the Conservation of Mass, as it is technically called. If the mass-values of bodies were assumed to vary in some regular manner with the time, with the size or proximity of neighbouring systems, or the like, the procedure of abstract mechanics would be more complicated than it proves to be on the simpler hypothesis of the constancy of such mass-values. But though actual facts conform to such an assumption, there is no necessity about it. Still less is there any justification for converting this principle of mass-conservation into an assertion concerning the mass of the universe either in respect of its quantity or its constancy. The epistemological character of mathematical mechanics as a purely descriptive apparatus would exclude these, as well as other real affirmations, from its scope. It would be as reasonable to expect from arithmetic a census of the separate bodies in the universe as to look to pure mechanics for an assurance that the mass of the universe is, as Helmholtz would have us regard it, an eternally unchangeable quantity. If there are any grounds for such a position at all, those grounds must lie either in a posteriori inferences from experience, that can never be more than probable, or in a priori reasoning of a non-mathematical kind.

But before a priori considerations can be brought to bear on such a point, mass must be identified with matter, and matter with substance. And this is precisely what we find in the plausible and summary argument of Mr. Spencer's First Principles. His crucial experimental proof is just that constancy of mass, gravitationally measured, which I have already mentioned. For, after citing several trivial instances, he clenches them with the remark: “Not, however, until the rise of quantitative chemistry, could the conclusion suggested by such experiences be reduced to a certainty.”14 Spite of this very restricted evidence for the conservation of mass as a simple and useful working hypothesis, we find Mr. Spencer concluding that “the form of our thought renders it impossible for us to have experience of Matter passing into non-existence,…that hence the indestructibility of Matter is in strictness an a priori truth”; albeit the ‘pseudo-thinking of undisciplined minds’ is ever leading them mistakenly to suppose they can really think ‘the absolutely unthinkable.’ Now the question is not at all whether we can or cannot conceive the universe to arise out of, or pass into, nothing; but simply what justification there may be for a certain absolute statement concerning that dynamical phenomenon we describe by help of the conception of Mass. When Mr. Spencer or some one else shall have shewn that what exists must exist as matter or not exist at all, and that all matter is necessarily ponderable matter, then, but not before, the old maxim, Ex nihilo nihil fit, and the appeal to the balance will be relevant to the question.

Quantity of mass is not necessarily identical with quantity of matter; and indeed, it seems obvious that, till matter is defined qualitatively, quantitive statements concerning it must be altogether precarious. Meanwhile, the prospects of a scientific definition of matter get more and more remote. The severely exact physicist of the Kirchhoff school, as we have seen, avoids the whole of this subject with disdain; while others with powerful scientific imagination like Faraday or Maxwell or Lord Kelvin, who pursue it eagerly, find themselves eluded in turn, and end, as Boltzmann says, by talking in parables.15 Yet such parables and analogies are of inestimable value, if only as a protest against the confident dogmatism of which Mr. Spencer is such a master. Consider, for example, Lord Kelvin's well-known vortex-atom theory of ponderable matter. According to his ideal presentation of it we are to imagine a perfect, i.e. absolutely frictionless fluid; then the rotational motion of portions of this fluid are what we know as ponderable matter; while the movements of these through the fluid are what we know as moving masses. In other words, our phenomenal matter is reduced to ‘non-matter in motion.’ This brilliant hypothesis (which has been accounted deserving of careful and minute attention by many of our leading physicists), suffices, even as it stands, to suggest what removes there may be between our physical, experiences and anything that must be conserved because its non-conservation is a priori inconceivable. But instead of taking this hypothesis as it stands, let us suppose, as the writers of the Unseen Universe do, that its ideal rigour is somewhat abated. Vortex rings in an absolutely perfect fluid would remain self-identical and undiminished forever; vortex rings in an indefinitely perfect fluid would so remain, not forever, but indefinitely long. But per contra, vortex rings in an indefinitely frictionless fluid could be originated through such processes as we find setting up vortices in the imperfect fluids about us; on a perfect fluid such processes would have no hold. Now, questions of theoretical simplicity and definiteness apart, there is no gainsaying the fact that there is no experimental need for assuming this ether-matter to be a perfect fluid. No balance is delicate beyond six decimal places, and our longest astronomical records are but ephemeral in comparison with cosmical ages. An ‘unbroken continuity’ is thus all that our experience requires, and this we have by regarding the hypothetical fluid of the vortex atoms as indefinitely perfect; and have not, if we regard it as absolutely so.16 Moreover, on the former alternative, we should be free to allow the possibility of ponderable matter coming to be here and ceasing to be there; the average amount in existence at once, either remaining stationary or else slowly altering, as is the case with the population of the globe, for example. Also we could entertain such a supposition without either flying in the face of any truth there is in what Mr. Spencer calls “the experimentally-established induction” that Matter is indestructible, or deserving his taunt of “not thinking at all, but merely pseudo-thinking.”

This hydro-kinetic theory of matter as a mode of motion and not a substance, is specially wholesome and instructive, if we compare it with the modern theory of heat as a mode of motion, that has replaced the older theory of caloric as a substance. We cannot conceive substance to be either produced or destroyed, Mr. Spencer will tell us. True and trite, we must allow. When therefore it was found that heat and mechanical work were mutually transformable, there was an end of the theory that heat was a substance. It is now possible to produce vortex rings, to show that their behaviour in many respects approximates strikingly to the behaviour of material particles, and that this approximation would be greater if the fluids at our disposal were less unlike the continuous and frictionless fluid supposed to fill all space. Thus, though man may never be able to make or unmake a material particle, Lord Kelvin's ingenious speculations may at least predispose us to believe in the thoroughly phenomenal character of all measurable masses, and, believing this, we are under no temptation to render absolute that relative constancy of such masses which our experience so far has disclosed.

How utterly unscientific it is to apply this principle of the conservation of mass to the entire universe is evident again when we reflect that it involves the further assertion that the universe is a finite system. Some recent writers on arithmetic talk of numbers that are at once infinite and complete, transfinite numbers as they are called. But it is obvious that there can be no scientific warrant for affirming such definite infinity of the universe, and there is certainly no empirical justification for affirming definite limits. No doubt what we see is limited; but to contend that we see no more, simply because there is no more to see, would be more illogical than it is to maintain that the bulk that may be beyond us must resemble the sample that we know. What we see is limited indeed in the sense of being finite, but it is not limited in the sense of being either constant or complete.

But now if the physicist were to ask the mathematician to devise for him a descriptive apparatus adapted to the movements of a material system in which the mass-values varied, the mathematician's first question would be: How do they vary? The physicist could not say. Innumerable forms of regular increase or decrease or of periodic alternation of the two are possible. Over against this bewildering variety the one definite supposition of constancy, in itself the simplest, is borne out by the very small fraction of the world that we can imperfectly measure. This seems to me how the case stands; and if it is, then it becomes plain that abstract dynamics affords as little ground for absolute statements about the magnitude or constancy of mass as for such statements concerning space or time. There are writers, however, who do not hesitate to rest this doctrine of the conservation of mass on that of the conservation of energy. But as this only means that in their opinion the latter doctrine cannot be true if the mass of the universe is not constant, such a plea is worthless unless there are independent reasons for maintaining that the energy of the universe is constant; and would not necessarily be true even then. The discussion of this important subject it will be best to defer till we have dealt with the application of abstract dynamics to the phenomena of molecular physics. To this I will ask your attention in the next lecture.

  • 1.

    It may be objected that such ‘simplest possible description’ is itself explanation, that in fact explanation is merely resolving the complex into the simple, and assimilating the less known to the better known. I admit this fully. But experience is not restricted to the range of exact science, and so far it is true that a fact is not fully explained if its cause is unknown. (Cf. below, Lecture 19.) Precisely in this lay the difficulty for such men as Huygens, Leibnitz, and Bernoulli of Newton's theory of gravitation. Newton only professed to ‘describe,’ but, as Lange tersely puts it: “These men could not separate the mathematics from the physics, and physically the doctrine of Newton was for them inconceivable.” And so it has remained till this day, although people are now accustomed to regard Newton's descriptive conception as if it were itself a physical cause.

  • 2.

    Cf. Newton's Preface to the Principia.

  • 3.

    Cf. Pemberton's translation, pp. 10 ff.

  • 4.

    Hypotheses of Dynamics, Phil. Mag., 1893, vol. 36, p. 237.

  • 5.

    Cf. Sigwart, Logic, § 88, 8; and Riehl, Der philosophische Kriticismus, Bd. II. i. pp. 92 ff.

  • 6.

    Cf. L. Lange, Die geschichtliche Entwicklung des Bewegungsbegriffes, 1886, p. 139.

  • 7.

    Properties of Matter, p. 92.

  • 8.

    Die Mechanik in ihrer Entwicklung, 2te Aufl., pp. 216 f. There is now an English translation of this most interesting book.

  • 9.

    But see the article by Professor MacGregor quoted above.

  • 10.

    Concepts and Theories of Modern Physics, p. 200.

  • 11.

    Philosophical Magazine, 1893, vol. 36, p. 40.

  • 12.

    Vorlesungen über mathematische Physik, p. 1.

  • 13.

    Cf. Helmholtz, Erhaltung der Kraft, p. 7.

  • 14.

    First Principles, § 52, stereo. ed., p. 173.

  • 15.

    Roger Cotes begins his Preface to the Principia by reducing natural philosophers to three classes: first, the Aristotelians, who attribute specific and occult qualities to things, and last, the experimentalists, who invent no hypotheses, among whom, of course, he places his ‘most celebrated author.’ The second reject the substantial forms of the peripatetics and lay down the principle that all matter is homogeneous. “But when,” he continues, “they assume to themselves a liberty of supposing at pleasure unknown figures and magnitudes, uncertain situations and motions of the parts; and moreover of supposing occult fluids, which freely pervade the pores of bodies, endowed with an all-powerful subtility, and agitated with occult motions; they then descend to visions, and neglect the true constitution of things.… Although they afterward proceed with the greatest accuracy from those principles [they] may be said to compose a fable, elegant, perhaps, and pleasing to the imagination, but still it is a fable.”

  • 16.

    Cf. Unseen Universe, second edition, p. 118.