An Enquiry Concerning the Principles of Natural Knowledge/Chapter 2

CHAPTER II

THE FOUNDATIONS OF DYNAMICAL PHYSICS

4. Newton’s Laws of Motion. 4.1 The theoretical difficulties in the way of the application of the philosophic doctrine of relativity have never worried practical scientists. They have started with the working assumptions that in some sense the world is in one euclidean space, that the permanent points in such a space have no individual characteristics recognisable by us, except so far as they are occupied by recognisable material or except in so far as they are defined by assigned spatial relations to points which are thus definitely recognisable, and that according to the purpose in hand either the earth can be assumed to be at rest or else astronomical axes which are defined by the aid of the solar system, of the stars, and of dynamical considerations deduced from Newton’s laws of motion.

4.2 Newton’s laws^[1] of motion presuppose the notions of mass and force. Mass arises from the conception of a passive quality of a material body, what it is in itself apart from its relation to other bodies; the notion of ‘force’ is that of an active agency changing the physical circumstances of the body, and in particular its spatial relations to other bodies. It is fairly obvious that mass and force were introduced into science as the outcome of this antithesis between intrinsic quality and agency, although further reflection may somewhat mar the simplicity of this outlook. Mass and force are measurable quantities, and their numerical expressions are dependent on the units chosen. The mass of a body is constant, so long as the body remains composed of the same self-identical material. Velocity, acceleration and force are vector quantities, namely they have direction as well as magnitude. They are thus representable by straight lines drawn from any arbitrary origin.

4.3 These laws of motion are among the foundations of science; and certainly any alteration in them must be such as to produce effects observable only under very exceptional circumstances. But, as is so often the case in science, a scrutiny of their meaning produces many perplexities.

In the first place we can sweep aside one minor difficulty. In our experience, a finite mass of matter occupies a volume and not a point. Evidently therefore the laws should be stated in an integral form, involving at certain points of the exposition greater elaboration of statement. These forms are stated (with somewhat abbreviated explanation) in dynamical treatises.

Secondly, Lorentz’s distinction between macroscopic equations and microscopic equations forces itself on us at once, by reason of the molecular nature of matter and the dynamical nature of heat. A body apparently formed of continuous matter with its intrinsic geometrical relations nearly invariable is in fact composed of agitated molecules. The equations of motion for such a body as used by an engineer or an astronomer are, in Lorentz’s nomenclature, macroscopic. In such equations even a differential element of volume is to be supposed to be sufficiently large to average out the diverse agitations of the molecules, and to register only the general unbalanced residuum which to ordinary observation is the motion of the body.

The microscopic equations are those which apply to the individual molecules. It is at once evident that a series of such sets of equations is possible, in which the adjacent sets are macroscopic and microscopic relatively to each other. For example, we may penetrate below the molecule to the electrons and the core which compose it, and thus obtain infra-molecular equations. It is purely a question as to whether there are any observed phenomena which in this way receive their interpretation.

The inductive evidence for the validity of Newton’s equations of motion, within the experimental limits of accuracy, is obviously much stronger in the case of the macroscopic equations of the engineer and the astronomer than it is in the case of the microscopic equations of the molecule, and very much stronger than in the case of the infra-microscopic equations of the electron. But there is good evidence that even the infra-microscopic equations conform to Newton’s laws as a first approximation. The traces of deviation arise when the velocities are not entirely negligible compared to that of light.

4.4 What do we know about masses and about forces? We obtain our knowledge of forces by having some theory about masses, and our knowledge of masses by having some theory about forces. Our theories about masses enable us in certain circumstances to assign the numerical ratios of the masses of the bodies involved; then the observed motions of these bodies will enable us to register (by the use of Newton’s laws of motion) the directions and magnitudes of the forces involved, and thence to frame more extended theories as to the laws regulating the production of force. Our theories about the direction and comparative magnitudes of forces and the observed motions of the bodies will enable us to register (by the use of Newton’s laws of motion) the comparative magnitudes of masses. The final results are to be found in engineers’ pocket-books in tables of physical constants for physicists, and in astronomical tables. The verification is the concordant results of diverse experiments. One essential part of such theories is the judgment of circumstances which are sufficiently analogous to warrant the assumption of the same mass or the same magnitude of force in assigned diverse cases. Namely the theories depend upon the fact of recognition.

4.5 It has been popular to define force as the product of mass and acceleration. The difficulty to be faced with this definition is that the familiar equation of elementary dynamics, namely,

$mf=P$

now becomes

$mf=mf$

It is not easy to understand how an important science can issue from such premisses. Furthermore the simple balancing of a weight by the tension of the supporting spring receives a very artificial meaning. With equal reason we might start with our theories of force as fundamental, and define mass as force divided by acceleration. Again we should be in equal danger of reducing dynamical equations to such identities as

$P/f=P/f$

Also the permanent mass of a bar of iron receives a very artificial meaning.

5. The Ether. 5.1 The theory of stress between distant bodies, considered as an ultimate fact, was repudiated by Newton himself, but was adopted by some of his immediate successors. In the nineteenth century the belief in action at a distance has steadily lost ground.

There are four definite scientific reasons for the adoption of the opposite theory of the transmission of stress through an intermediate medium which we will call the ‘ether.’ These reasons are in addition to the somewhat vague philosophic preferences, based on the disconnection involved in spatial and temporal separation. In the first place, the wave theory of light also postulates an ether, and thus brings concurrent testimony to its existence. Secondly, Clerk Maxwell produced the formulae for the stresses in such an ether which, if they exist, would account for gravitational, electrostatic, and magnetic attractions. No theory of the nature of the ether is thereby produced which in any way explains why such stresses exist; and thus their existence is so far just as much a disconnected assumption as that of the direct stresses between distant bodies. Thirdly, Clerk Maxwell’s equations of the electromagnetic field presuppose events and physical properties of apparently empty space. Accordingly there must be something, i.e. an ether, in the empty space to which these properties belong. These equations are now recognised as the foundations of the exact science of electromagnetism, and stand ona level with Newton’s equations of motion. Thus another testimony is added to the existence of an ether.

Lastly, Clerk Maxwell’s identification of light with electromagnetic waves shows that the same ether is required by the apparently diverse optical and electromagnetic phenomena. The objection is removed that fresh properties have to be ascribed to the ether by each of the distinct lines of thought which postulate it.

It will be observed that gravitation stands outside this unification of scientific theory due to Maxwell’s work, except so far that we know the stresses in the ether which would produce it.

5.2 The assumption of the existence of an ether at once raises the question as to its laws of motion. Thus in addition to the hierarchy of macroscopic and microscopic equations, there are the equations of motion for ether in otherwise empty space. The à priori reasons for believing that Newton’s laws of motion apply to the ether are very weak, being in fact nothing more than the inductive extension of laws to cases widely dissimilar from those for which they have been verified. It is however a sound scientific procedure to investigate whether the assumed properties of ether are explicable on the assumption that it is behaving like ordinary matter, if only to obtain suggestions by contrast for the formulation of the laws which do express its physical changes.

The best method of procedure is to assume certain large principles deducible from Newton’s laws and to interpret certain electromagnetic vectors as displacements and velocities of the ether. In this way Larmor has been successful in deducing Maxwell’s equations from the principle of least action after making the necessary assumptions. In this he is only following a long series of previous scientists who during the nineteenth century devoted themselves to the explanation of optical and electromagnetic phenomena. His work completes a century of very notable achievement in this field.

5.3 But it may be doubted whether this procedure is not an inversion of the more fundamental line of thought. It will have been noted that Newton’s equations, or any equivalent principles which are substituted for them, are in a sense merely blank forms. They require to be supplemented by hypotheses respecting the nature of the stresses, of the masses, and of the motions, before there can be any possibility of their application. Thus by the time that Newton’s equations of motion are applied to the explication of etherial events there is a large accumulation of hypotheses respecting things of which we know very little. What in fact we do know about the ether is summed up in Maxwell’s equations, or in recent adaptations of his equations such as those due to Lorentz. The discovery of electromagnetic mass and electromagnetic momentum suggests that, for the ether at least, we gain simpler conceptions of the facts by taking Maxwell’s equations, or the Lorentz-Maxwell equations, as fundamental. Such equations would then be the ultimate microscopic equations, at least in the present stage of science, and Newton’s equations become macroscopic equations which apply in certain definite circumstances to etherial aggregates. Such a procedure does not prejudge the debated theory of the purely electromagnetic origin of mass.

5.4 The modern theory of the molecule is destructive of the obviousness of the prejudgment in favour of the traditional concepts of ultimate material at an instant. Consider a molecule of iron. It is composed of a central core of positive electricity surrounded by annular clusters of electrons, composed of negative electricity and rotating round the core. No single characteristic property of iron as such can be manifested at an instant. Instantaneously there is simply a distribution of electricity and Maxwell’s equations to express our expectations. But iron is not an expectation or even a recollection. It is a fact; and this fact, which is iron, is what happens during a period of time. Iron and a biological organism are on a level in requiring time for functioning. There is no such thing as iron at an instant; to be iron is a character of an event. Every physical constant respecting iron which appears in scientific tables is the register of such a character. What is ultimate in iron, according to the traditional theory, is instantaneous distributions of electricity; and this ultimateness is simply ascribed by reason of a metaphysical theory, and by no reason of observation.

5.5 In truth, when we have once admitted the hierarchy of macroscopic and microscopic equations, the traditional concept is lost. For it is the macroscopic equations which express the facts of immediate observation, and these equations essentially express the integral characters of events. But this hierarchy is necessitated by every concept of modern physics — the molecular theory of matter, the dynamical theory of heat, the wave theory of light, the electromagnetic theory of molecules, the electromagnetic theory of mass.

6. Maxwell’s Equations.^[2] 6.1 A discussion of Maxwell’s equations would constitute a treatise on electromagnetism. But they exemplify some general considerations on physical laws.

These equations (expressed for an axis-system α) . involve for each point of space and each instant of time the vector quantities $(F_{\alpha },G_{\alpha },H_{\alpha })$, $(L_{\alpha },M_{\alpha },N_{\alpha })$ and $(u_{\alpha },v_{\alpha },w_{\alpha })$, namely the electric and magnetic ‘forces’ and the velocity of the charge of electricity. Now a vector involves direction; and direction is not concerned with what is merely at that point. It is impossible to define direction without reference to the rest of space; namely, it involves some relation to the whole of space.

Again the equations involve the spatial differential operators ${\frac {\partial }{\partial x_{\alpha }}}$, ${\frac {\partial }{\partial y_{\alpha }}}$, ${\frac {\partial }{\partial z_{\alpha }}}$, which enter through the symbols $curl_{\alpha }$ and $div_{\alpha }$; and they also involve the temporal differential operator ${\frac {\partial }{\partial t_{\alpha }}}$. The differential coefficients thus produced essentially express properties in the neighbourhood of the point $(x_{\alpha },y_{\alpha },z_{\alpha })$ and of the time $t_{\alpha }$, and not merely properties at $(x_{\alpha },y_{\alpha },z_{\alpha },t_{\alpha })$. For a differential coefficient is a limit, and the limit of a function at a given value of its argument expresses a property of the aggregate of the values of the function corresponding to the aggregate of the values of the argument in the neighbourhood of the given value.

This is essentially the same argument as that expressed above in 1.2 for the particular case of motion. Namely, we cannot express the facts of nature as an aggregate of individual facts at points and at instants.

6.2 In the Lorentz-Maxwell equations [cf. Appendix II] there is no reference to the motion of the ether. The velocity $(u_{\alpha },v_{\alpha },w_{\alpha })$ which appears in them is the velocity of the electric charge. What then are the equations of motion of the ether? Before we puzzle over this question, a preliminary doubt arises. Does the ether move?

Certainly, if science is to be based on the data included in the Lorentz-Maxwell equations, even if the equations be modified, the motion of the ether does not enter into experience. Accordingly Lorentz assumes a stagnant ether: that is to say, an ether with no motion, which is simply the ultimate entity of which the vectors $(F_{\alpha },G_{\alpha },H_{\alpha })$ and $(L_{\alpha },M_{\alpha },N_{\alpha })$ express properties. Such an ether has certainly a very shadowy existence; and yet we cannot assume that it moves, merely for the sake of giving it something to do.

6.3 The ultimate facts contemplated in Maxwell’s equations are the occurrences of $\rho _{\alpha }$ (the volume-density of the charge), $(u_{\alpha },v_{\alpha },w_{\alpha })$, $(F_{\alpha },G_{\alpha },H_{\alpha })$ and $(L_{\alpha },M_{\alpha },N_{\alpha })$ at the space-time points in the neighbourhood surround- ing the space-time point $(x_{\alpha },y_{\alpha },z_{\alpha },t_{\alpha })$. But this is merely to say that the ultimate facts contemplated by Maxwell’s equations are certain events which are occurring throughout all space. The material called ether is merely the outcome of a metaphysical craving. The continuity of nature is the continuity of events; and the doctrine of transmission should be construed as a doctrine of the coextensiveness of events with space and time and of their reciprocal interaction. In this sense an ether can be admitted; but, in view of the existing implication of the term, clearness is gained by a distinction of phraseology. We shall term the traditional ether an ‘ether of material’ or a ‘material ether,’ and shall employ the term ‘ether of events’ to express the assumption of this enquiry, which may be loosely stated as being ‘that something is going on everywhere and always.’ It is our purpose to express accurately the relations between these events so far as they are disclosed by our perceptual experience, and in particular to consider those relations from which the essential concepts of Time, Space, and persistent material are derived. Thus primarily we must not conceive of events as in a given Time, a given Space, and consisting of changes in given persistent material. Time, Space, and Material are adjuncts of events. On the old theory of relativity, Time and Space are relations between materials; on our theory they are relations between events. Page:An Enquiry Concerning the Principles of Natural Knowledge.djvu/41 Page:An Enquiry Concerning the Principles of Natural Knowledge.djvu/42 Page:An Enquiry Concerning the Principles of Natural Knowledge.djvu/43 stands for what is called the vector product of the two vectors, namely the vector

$({{Y_{\alpha }}^{\prime }}{Z_{\alpha }}-{{Z_{\alpha }}^{\prime }}{Y_{\alpha }},{{Z_{\alpha }}^{\prime }}{X_{\alpha }}-{{X_{\alpha }}^{\prime }}{Z_{\alpha }},{{X_{\alpha }}^{\prime }}{Y_{\alpha }}-{{Y_{\alpha }}^{\prime }}{X_{\alpha }})$

It is evident that ${curl_{\alpha }}(X_{\alpha },Y_{\alpha },Z_{\alpha })$ can be expressed in the symbolic form

${\bigg [}{\bigg (}{\frac {\partial }{\partial x_{\alpha }}},{\frac {\partial }{\partial y_{\alpha }}},{\frac {\partial }{\partial z_{\alpha }}}{\bigg )}.(X_{\alpha },Y_{\alpha },Z_{\alpha }){\bigg ]}$

The vector equation

$(X_{\alpha },Y_{\alpha },Z_{\alpha })=({X_{\alpha }}^{\prime },{Y_{\alpha }}^{\prime },{Z_{\alpha }}^{\prime })$

is an abbreviation of the three equations

$X_{\alpha }={X_{\alpha }}^{\prime },Y_{\alpha }={Y_{\alpha }}^{\prime },Z_{\alpha }={Z_{\alpha }}^{\prime }$

Let $(F_{\alpha },G_{\alpha },H_{\alpha })$ be the electric force at $(x_{\alpha },y_{\alpha },z_{\alpha },t_{\alpha })$, and let $(L_{\alpha },M_{\alpha },N_{\alpha })$ be the magnetic force at the same point and time. Also let $\rho _{\alpha }$ a be the volume density of the electric charge and $(u_{\alpha },v_{\alpha },w_{\alpha })$ its velocity; and let $(P_{\alpha },Q_{\alpha },R_{\alpha })$ be the ponderomotive force: all equally at $(x_{\alpha },y_{\alpha },z_{\alpha },t_{\alpha })$. Finally let $c$ be the velocity of light in vacuo.

Then Lorentz’s form of Maxwell’s equations is

(1) ${{\text{div}}_{\alpha }}(F_{\alpha },G_{\alpha },H_{\alpha })=\rho _{\alpha }$

(2) ${{\text{div}}_{\alpha }}(L_{\alpha },M_{\alpha },N_{\alpha })=0$

(3) ${curl_{\alpha }}(L_{\alpha },M_{\alpha },N_{\alpha })=-{\frac {1}{c}}{\bigg \{}{\frac {\partial }{\partial t_{\alpha }}}(F_{\alpha },G_{\alpha },H_{\alpha })+{\rho _{\alpha }}(u_{\alpha },v_{\alpha },w_{\alpha }){\bigg \}}$

(4) ${curl_{\alpha }}(F_{\alpha },G_{\alpha },H_{\alpha })=-{\frac {1}{c}}{\frac {\partial }{\partial t_{\alpha }}}(L_{\alpha },M_{\alpha },N_{\alpha })$

(5) $(P_{\alpha },Q_{\alpha },R_{\alpha })=(F_{\alpha },G_{\alpha },H_{\alpha })+{\frac {1}{c}}[(u_{\alpha },v_{\alpha },w_{\alpha }).(L_{\alpha },M_{\alpha },N_{\alpha })]$

It will be noted that each of the vector equations (3), (4), (5) stands for three ordinary equations, so that there are eleven equations in the five formulae.

Notes

1. Cf. Appendix I to this chapter.
2. Cf. Appendix II to this chapter.