Mary B. Hesse - Acctions in Distance in Classical Physics

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Action at a Distance in Classical Physics Author(s): Mary B. Hesse Source: Isis, Vol. 46, No. 4 (Dec., 1955), pp. 337-353Published by: on behalf of The University of Chicago Press The History of Science SocietyStable URL: http://www.jstor.org/stable/227576Accessed: 11-03-2015 23:38 UTC

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Action at a Distance

in Classical Physics

By Mary B. Hesse *

1

THE scholastic axiom that "matter cannot act where it is not" is one of the very general metaphysical principles found in science before the seven-

teenth century which retain their relevance for scientific theory even when the metaphysics itself has been discarded. Other such principles have been fruitful in the development of physics: for example, the "conservation of motion" stated by Descartes and Leibniz, which was generalized and given precision in the nineteenth century as the doctrine of the conservation of energy; and the principle that nature works by the shortest route, which was hailed by Mauper- tuis as a metaphysical principle "so wise, so worthy of supreme Being," and which led directly to variational methods in optics and dynamics.

We may regard these metaphysical statements as generalizations from familiar experience, or from a familiar stock of ideas, applied analogically to the fundamental structure of nature. In this they are like all scientific hypotheses, but they lack the precision of a good hypothesis, and are capable of various interpretations depending on the type of theory in which they are used; also they are often not held tentatively, like hypotheses, but are unconscious presuppositions of science into which all the facts are made to fit. It may be of interest to sketch part of the historical background of one of these principles, namely, the alleged impossibility of action at a distance, and to try to reinter- pret some of the controversies that have arisen round it, not in terms of rival ontological theories, but in terms of the choice of suitable models or analogues for the description of physical phenomena.

It has been characteristic of physics that it is only those actions which produce or tend to produce motion that are considered as candidates for fundamental explanations. Actions involving change of other properties, such as il- lumination or temperature, are explained in terms of motion, and not the re- verse. In other words mechanics has traditionally provided the basic categories of physics. The types of action to be admitted in mechanics were further restricted at the beginning of the modern scientific period by the decision to explain the change of motion of bodies only by communication from outside, and not by any innate power or striving within the bodies themselves. Collingwood credits Kepler with "the momentous step of proposing that in treating of physics the word anima should be replaced by the word vis: in other words, that the conception of vital energy producing qualitative changes should be replaced by that of mechanical energy . . . . producing quantitative changes." 1

*University of Leeds, England. 1 The Idea of Nature (Oxford, 1945), p. ICI.

337



338 MARY B. HESSE

Galileo was the first to exploit this principle systematically in the study of the motion of inert matter; in the physics of Descartes mechanical categories were the only ones admitted in explaining the behavior of material substance, as opposed to mind; and from their time there was a gradual elimination of the analogies from living beings and the teleological categories which had pervaded Aristotelian and Renaissance physics. Certain features of animistic physics lingered on in seventeenth-century speculations upon the vapors and emanations invoked to explain magnetic and electric actions, and even in the "etherial spirits" and "active principles" discussed by Newton in letters to Oldenburg and Boyle. In earlier writers such as Telesio, Bruno and Gilbert these emanations were not entirely material in the sense in which "matter" was later sharply dis- tinguished from "spirit," but they had something of the nature of spirit or soul, conceived as a fine vapor pervading bodies. At first the vapors were supposed to have some capacity for initiating action, for instance in drawing iron towards a magnet, but this capacity ceased to be required as time went on, and, if any detailed account of the functioning of the vapor, or aether,2 were attempted, it was in terms of the mechanical laws of ordinary matter.

In the mechanical theory of matter which came to dominate natural philosophy it was assumed that the basic constituents of nature are substances having most of the properties of matter as ordinarily observed, namely, position and extension in space, persistence through time, impenetrability and so on. Within this framework various choices between different basic concepts and different types of mechanical action presented themselves, namely, whether matter is ultimately continuous or discrete, whether there is a material aether, whether there is void, and whether action at a distance may be admitted. Fundamental actions between parts of matter and aether were conceived in three different ways: as impacts, as actions in a continuous medium, and as actions at a distance. Each of these exhibits itself to common-sense observation in a familiar type of mechanical process which may be called the physical model 3 of the action, and each gave rise during the seventeenth and eighteenth centuries to a characteristic type of mathematical theory. Empirical and mathematical study of action by impact culminated in Newton's laws of elastic and inelastic impact in his Principia Mathematica. Action in a continous medium was first studied in terms of the communication of action through fluids by direct pressure or by wave motion, the theory of which was worked out in the hydrody- namics of Newton, the Bernouillis, and others, and later, in the first half of the nineteenth century, an alternative physical and mathematical model was provided by the development of the theory of continuous elastic solids.4 Finally,

' "Aether" may be used as a generic term for all those "subtle substances" postulated for various reasons by the seventeenth-century writers; by Descartes, for instance, because he denied the existence of void, also by Descartes, Huyghens and Newton, to account for certain of the motions of gross matter. The distinction between matter and aether sharpened as the properties of matter came to be better understood and as Newtonian mechanics was estab- lished, and weight became the chief distinguish-

ing property of matter, aether being one of the "imponderable fluids" introduced in several branches of physics in the eighteenth century.

'See the author's "Models in Physics," Brit- ish Journal for the Philosophy of Science, I953,

4: I98. 'It should be remarked that in the mathe-

matics of these theories matter is regarded as continuous, and no account is taken of small- scale atomicity. The application of the theories to matter which is regarded as molecular is



ACTION AT A DISTANCE 339

action at a distance was studied in connection with falling bodies, the solar system, and electric and magnetic attractions, and was given mathematical shape in Newton's theory of central forces, including gravitation.

Until recently all the theories of mechanical motions have been derivatives of one or more of these three models. Much of the progress of theoretical physics can be described in terms of attempts to extend the application of each model to new fields, and at the same time to reduce two of the models to the remaining one in order to economize hypotheses. For instance, action by impact is extended to the behaviour of gases, continuous action to the phenomena of light and electromagnetic radiation, and action at a distance to the interior of the atom. On the other hand, the view taken of the nature of matter has at one time required impact or continuous action to be regarded ultimately as action at a distance, and has at another time required electric and magnetic attraction to be continuous action in an aether.

During the seventeenth and early eighteenth centuries action at a distance was regarded with suspicion. Leibniz attacked it in his long correspondence with Samuel Clarke, describing it as a means of communication which is "invisible, intangible, not mechanical." Clarke might as well have added, Leibniz goes on, that it is "inexplicable, unintelligible, precarious, groundless and unex- ampled. . . . Of which sorts of things, the author seems to have still a good stock in his head. . . . 'Tis a chimerical thing, a scholastic occult quality." 5

This indicates the grounds of criticism: those who introduced action at a distance were accused of returning to the Aristotelian habit of postulating an ad hoc quality for every new phenomenon, without showing that the quality explained the phenomenon in any way by relating it to other processes of nature. When Newton's disciples were driven to speak of the means by which bodies attract each other as "invisible, intangible and non-mechanical," they seemed to their contemporaries to be surrendering to the immaterial influences and sym- pathies which had been banished from physics so recently and with such difficulty.

The new orthodoxy was Cartesianism, the first thorough-going attempt to base physics on nothing but substance and motion. The physical model developed by Descartes was that of the motion of a continuous fluid medium dif- ferentiated into particles of various sizes forming gross matter, and a subtle, insensible aether which fills the rest of space, allowing no void. Motion in such a universe had to be vortical, and action at a distance was not admitted. Descartes gave explanations in these terms of phenomena ranging from the planetary orbits to chemical reactions and the circulation of the blood, explanations which are ingenious and often picturesque, but he did not have the mathematical equipment to develop a complete hydrodynamical theory which would have enabled him to test the continuous fluid hypothesis against ob-

saved by some such rubric as "Consider an ele- ment which is small compared to macroscopic dimensions but large compared to single molecules," so that the effect of molecularity is smoothed out. But this particular application is irrelevant to the point being made here, namely,

that there exists a mathematical theory which can deal with action in a continuous medium, and which can be taken as a model for all such action.

'The Works of Samuel Clarke (London, 1738), Leibniz's fifth paper, Vol. IV, 668.



340 MARY B. HESSE

servation. Having constructed such a theory Newton was able to show easily enough that a vortex theory of the solar system does not account for the planetary motions.6

The position of Newton himself with regard to the alternative models for action is ambiguous, for although his theory of gravitation is the prototype of action at a distance theories, he himself was of the opinion that it might ultimately be possible to explain action at a distance in terms of impact. Mathe- matically the theory of gravitation is a special case of the theory of central forces (or "centripetal" forces as Newton called them), and this theory is an immediate consequence of Newton's definition of centripetal force 7 as "that by which bodies are drawn or impelled, or in any way tend, towards a point as to a centre." Newton gives as examples gravity, magnetism, and the force on a stone whirled in a sling. His definition abstracts from the physical means whereby the force is exerted, and concerns itself only with the resulting tend- ency of the body to move towards the centre:

. . . the reader is not to imagine, that . . . I anywhere take upon me to define the kind, or the manner of any action, the causes or the physical reason thereof, or that I attribute forces, in a true and physical sense, to certain centres (which are only mathematical points); when at any time I happen to speak of centres as attracting, or as endowed with attractive powers.8

Newton held that this theory of forces does not imply anything about the physical means by which they are produced. The attractive force was derived immediately from phenomena, since if the mass and acceleration of a moving body are given, the magnitude of the force acting upon it can be found from the laws of motion, and in this sense the theory of gravitation was not a hypothesis, but a manner of describing phenomena, and a necessary pre- liminary to the attempt to find the physical cause of attraction. Newton therefore denied that attraction was in any sense an "occult quality" of bodies, like the qualities postulated by the Aristotelians, for attractive forces could be used to derive the motions of bodies, whereas the Aristotelian qualities were mere names which explained nothing. In Newton's view, and to a greater extent in the view of his disciple Cotes, no explanation of attracting forces was necessary to a physical system, but in some of his writings 9 Newton allows himself to speculate upon the possible effects of a "subtle substance" whose action upon gross matter might produce in bodies the appearance of attracting one another. And in a letter to Bentley dated I693 he definitely asserts that it is absurd to suppose that gravity is innate and acts without a medium, either material or immaterial.10

On the nature and precise properties of this "material or immaterial" medium, however, Newton was unable to do more than make tentative suggestions. The suggestion that the medium might not have the properties of ordinary matter could not be taken into an increasingly mathematical physics

'Philosophiae Naturalis Principia Mathema- tica (London, I687), Book II, Section ix.

7Ibid., p. 3, definition v. 8Ibid., p. 8, definition viii. 'Letter to Boyle, Correspondence of Scien-

tific Men of the Seventeenth Century, ed. Rigaud (Oxford, I841), vol. II, p. 407; letters to Bentley, The Works of Richard Bentley (London, I838), vol. III.

'0lbid., p. 2I2.



ACTION AT A DISTANCE 34I

until the notion of "immaterial" was divested of its association with spirits, vapors, emanations and the like, which still lingered from animistic physics, and given a form which could be expressed mathematically. This had to wait for the nineteenth century.

As physical science consolidated its position and as empirical philosophy became more influential, metaphysical objections to action at a distance lost most of their plausibility. In 1763 Boscovitch could so far turn Leibnizian arguments against their author as to assert boldly that action at a distance was a necessary consequence of Leibniz's own "law of continuity." He concluded that the ultimate particles of matter cannot be finite, because that would involve a discontinuous change of density at their boundaries; matter therefore consists of points of no extent having inertia and exerting forces on one another depending on their mutual distances. In this way are explained gravitational, electric and magnetic attraction, the cohesion and stability of aggre- gates of point masses, and the repulsive force exhibited in impact, all in terms of forces acting at a distance.

Boscovitch claimed that these forces were not mysterious qualities, but simply the ideas of propensity to approach and to recede:

. . .the various motions that arise from forces of this kind, such as when one body collides with another body, when one part of a solid is seized and another part follows it, when the particles of gases, and of springs, repel one another, when heavy bodies descend, these motions, I say, are of everyday occurrence before our eyes. . . . In all of these there is nothing mysterious; on the contrary they all tend to make the law of forces of this kind perfectly plain."

2

By the beginning of the nineteenth century there were, generally speaking, two fundamental theories in physics, one accounting for the propagation of light by postulating an aether, and the other accepting action at a distance as the basis of gravitation and electric and magnetic attractions, though the possibility of an ultimate explanation of these by aether action was not denied. The mathematical theory of attractions was developed by Laplace and Poisson, who treated the space surrounding centres of force as the field of a potential function, so that at any point a value could be given to the force that would be exerted on a body placed at that point. This was merely a mathematical de- vice, and not an attempt to assign properties to any aetherial medium filling the field, so that, physically speaking, action was still regarded as exerted at a distance by the centres of force. It was Faraday who first challenged the ade- quacy of this; he was not satisfied with pure action at a distance, but wished to picture the physical events going on in the intervening medium more con- cretely than was allowed for by the potential functions.

Faraday's contributions to theoretical physics lay in his ability to think in physical pictures rather than in mathematical terms. He freely admits that he is not competent to judge such work as Poisson's treatment of electric and

1' Theoria Philosophiae Naturalis (first ed. Vienna, I763), Eng. trans. Child (Chicago, 1922), P. 95.



342 MARY B. HESSE

magnetic field theory, but later, in the hands of Kelvin and Maxwell, his own pictures of "lines of force" emanating from charged conductors and from magnets became the basis of further mathematical developments. Faraday's use of pictorial representations led him to discussion and reinterpretation of the fundamental concepts of action by contact and action at a distance.

It was in regard to electric induction that Faraday first became convinced that the action was propagated through a medium and not at a distance. His experimental researches led to three conclusions which seemed to point to the existence of an active medium:

(i) The induction of electric charge between conductors across an insulating medium depends quantitatively on the nature of the insulator.

(ii) If the insulator is cut and the parts separated, opposite charges appear on the two separated surfaces.12

(iii) The lines of induction are curved, as illustrated by the spark of a discharge, and by experiments showing how the force on a charged ball due to a charged insulator is affected by the presence and shape of intervening conductors, which may make induction "turn a corner." 13

Faraday concluded that the insulating medium propagates the electric induction by means of its own particles, each of which is itself a conductor and becomes polarized, one side having a negative charge and the other a positive charge. This action takes place between contiguous particles along curved lines, and Faraday thinks that the fact that the lines are curved is strong proof that induction is "an action of continuous particles affecting each other in turn, and not an action at a distance." 14 Elsewhere he speaks of action between "contiguous particles" and explains more carefully what he means by "contiguous":

The word contiguous is perhaps not the best that might have been used here and elsewhere; for as particles do not touch each other it is not strictly correct.... By contiguous particles I mean those which are next.",

So that ultimately it seems that even this action is at a distance if regarded on the atomic scale. Faraday is explicit about this in correspondence with Dr. R. Hare of Pennsylvania."6 Dr. Hare had objected that rarefication of the air between conductors does not affect the transmission of electric induction, and that therefore the material medium cannot be essential. He suggested that an imponderable matter should be postulated, so that the polarization of this matter between conductors would "connect the otherwise imperfect chain of causes." Faraday replies that his use of "contiguous" includes a vacuum in which the particles of air may be separated by distances of the order of half an inch, but he will not commit himself here on the hypothesis of an imponderable aether. Two years earlier, however, he had had the same idea: "May inductive action be transmitted by other particles than those of ponderable matter, as by the particles of the supposed ether?" 17 Induction could not

"Diary (London, I933), vol. III, pp. 72ff. " Experimental Researches in Electricity

(London, 1839), vol. I, para. 1224. 14Diary, vol. III, p. 88.

5 Experimental Researches, vol. I, para. II64, fn.

1Ibid., vol. II, pp. 251ff. 17Diary, vol. III, p. 2I3.




then take place in a perfect vacuum, devoid of air and aether, but there was of course no means of checking that hypothesis as long as there was no independent means of detecting the supposed aether. The only thing in favor of it at the time was a prejudice against action at a distance.

It is noticeable, however, that the meaning of "action by contact" has come to be modified in Faraday's work. It is neither action by impact nor action in a continuous medium as these have been defined above, but action at a distance between particles at "insensible" distances apart. The question as to whether transmission of induction in a vacuum requires the postulation of an aether to save the principle of action by contact is left undecided. Although this new conception of action by contact gives perhaps a more satisfying physical picture of the chain of causality, its introduction makes no difference to the mathematical techniques employed, which are still essentially those of central forces varying according to the inverse-square law. Physically, action at a distance on the atomic scale is retained by Faraday, because he regards each particle of the intervening medium as being itself a conductor, and the properties of insulators as arising from the assumption that the conducting particles are not in contact with each other. If they were in contact, there would be no insulators.

Maxwell, however, interpreted Faraday's work as a replacement of concepts of action at a distance by continuous action in a different sense:

Faraday, in his mind's eye, saw lines of force traversing all space where the mathe- maticians saw centres of force attracting at a distance: Faraday saw a medium where they saw nothing but distance: Faraday sought the seat of the phenomena in real actions going on in the medium, they were satisfied that they had found it in a power of action at a distance impressed on electric fluids.'8

And again:

This [conception of lines of force] is quite a new conception of action at a distance, reducing it to a phenomenon of the same kind as that action at a distance which is exerted by means of tensions of ropes and the pressure of rods.'9

This brings us to a consideration of Faraday's magnetic lines of force, and to the question of how far his theory of electromagnetism is based on the physical model of stresses and strains in an elastic medium. If this were the physical model for magnetic action, then Faraday would have reduced magnetic action at a distance to action in a continuous medium, just as his contemporaries were attempting to describe the transmission of light by theories of an elastic aether. However, I hope to show that a continous elastic medium can be said to be a model in Faraday's work only in a very general sense.

Faraday's attitude towards speculative hypotheses was similar to that of Newton. He is not prepared to make any definite statement about the ultimate nature of the lines of force, although he is sometimes prepared, with apolo- gies,20 to make suggestions about it. It is only necessary to regard lines of

Electricity and Magnetism (Oxford, i88i), vol. I, p. X.

'Scientific Papers, vol. II, p. 320. ' See his paper "On the physical character of

the lines of magnetic force," Experimental Re- searches, vol. III, p. 407, and "Thoughts of Ray Vibrations," an address to the Royal Institu- tion (ibid., p. 452).



344 MARY B. HESSE

force as a pictorial representation of the direction of the force at any point of a magnetic field:

The term line of magnetic force is intended to express simply the direction of the force in any given place, and not any physical idea or notion of the manner in which the force may be there exerted; as by actions at a distance, or pulsations, or waves, or a current, or what not.21

The lines of force are simply what would be represented by iron filings scattered in the field.

Now Faraday uses a terminology in speaking about the lines of force which is derived from the idea of a bundle of elastic strings stretched under tension from point to point of the field. Thus he speaks of "tension" and "the number of lines" cut by a body moving in the field. Remembering his discussion about contiguous particles of a dielectric medium, one must think of the strings as stretching from one particle of the medium to the next in a straight line, the distance between particles being so small that the line appears as a smooth curve. How seriously does he take this model? Certainly the bundle of elastic strings is nothing like those one can buy at the store. The "number of lines" does not refer to a definite number of discrete material entities, but to the amount of force exerted over a given area in the field. It would not make sense to assign points through which a line passes and points which are free from a line. The field of force is continuous. Again, a conducting wire can cut these lines and they remain unbroken. There is no need to labor the point that the lines of force are not to be regarded as material entities. But the terminology of elastic strings may indicate that what is being described is a continuous elastic medium whose stresses and strains produce magnetic action; in other words, Faraday may be using a mathematical model of the elastic aether. This, however, is clearly not the case, for there is no development of the mathematical theory of elasticity in Faraday's work.

The physical model that Faraday is using is simply the pattern into which iron filings fall when they are scattered in a magnetic field. In other words his lines of force are not in themselves an attempt to explain the nature of magnetic action, but only a convenient way of picturing the experimental facts about the forces acting in the neighborhood of magnets and electric currents. In addition to the elastic string terminology Faraday uses phrases like "conducting power," which express an analogy between magnetic force and electric currents. But here again all that is intended is a convenient way of stating the fact that iron concentrates the lines of force by saying that iron is a better conductor of magnetic force than is air. The analogy cannot necessarily involve the further statement that something is travelling along the lines of force as charges travel along lines of current flow, although it may suggest that this is a profitable hypothesis. In modern terminology the word "tension" has been retained in speaking of lines of force, while the phrase "conducting power" has not, but both are "dead metaphors," in the sense that they do not involve any

21Experimental Researches, vol. III, p. 402; also pp. 328ff.




important physical analogy between elastic strings and lines of force, or between electric currents and lines of force. There are important mathematical analogies which were worked out later on the basis of Faraday's suggestions, and which we shall consider in connection with the work of Kelvin and Max- well.

When Faraday speaks about action at a distance and continuous action, he is thinking physically rather than mathematically. Later on, as we shall see, these concepts have to be expressed differently to take account of new mathematical theories, but, although Faraday paved the way for this restatement, his own discussions of the ultimate nature of action are still in terms of the traditional medium, which may be continuous or discrete, and the traditional atoms, which may be "billiard balls" or point-centers of force. The representation in terms of lines of force does not in itself commit him to any decision between these alternatives, and indeed he sees that this is one of the advantages of the representation. The arguments he uses about the nature of action are based on different physical considerations, and as they raise the question of action at a distance with a new clarity and a new relevance to the physics of Faraday's time, I shall describe them in some detail.

We have seen how the properties of electric induction led Faraday to the view that this is not an action at sensible distances, but requires the interven- tion of a material medium. He is concerned to discover whether the same can be said of magnetic action:

How the magnetic force is transferred through bodies or through space we know not: - whether the result is merely action at a distance, as in the case of gravity; or by some intermediate agency, as in the cases of light, heat, the electric current, and (as I believe) static electric action.22

In a paper entitled "On the physical character of the lines of magnetic force," 23 in which Faraday remarks that he is leaving "the strict line of reasoning" and entering upon "a few speculations," he suggests some criteria by which different kinds of action may be recognized:

(i) Can transmission of action be affected by changes in the intervening medium, as regards, for instance, a bending of the lines or polarity effects?

(ii) Does the transmission take time? (iii) Does it depend upon the "receiving" end?

These questions are answered with respect to gravity, radiation, and electric and magnetic force.

First, with respect to gravity: (i) Nothing in the intervening medium affects a line of gravitational force be-

tween two particles. The line is straight, no matter what other particles may be in the field, and the action between any pair of particles is independent of that between any other pair.

(ii) It seems impossible to prove whether or not gravity takes time. "If it did, 2 Ibid., p. 330. ' ibid., p. 407.



346 MARY B. HESSE

it would show undeniably that a physical agency existed in the course of the line of force." 24

(iii) The action of gravity is dependent upon the mass of both reacting particles and their distance apart.

So gravity presents us with the simplest case of attraction; and appearing to have no relation to any physical process by which the power of the particles is carried on between them, seems to be a pure case of attraction or action at a distance, and offers therefore the simplest type of the cases which may be like it in that respect.24

Second, with regard to radiation: (i) Lines of radiation are affected by the properties of the intervening medium

both in curvature and in transverse orientation about their axis (polarity). (ii) They require time for their propagation.

(iii) They are not dependent upon a second reaction particle.

Here "we obtain the highest proof, that though nothing ponderable passes, yet the lines of force have a physical existence independent, in a manner, of the body radiating, or of the body receiving the rays." 24

Third, with regard to electric induction: (i) Lines of electric induction are affected by the material medium, but it is

not certain whether, in a vacuum, they would be straight like those of gravity, or curved. No condition of polarity has been observed.

(ii) No time has been shown to be required for their propagation. (iii) A second reacting particle is required.

Fourth, with regard to electric current: (i) Current is affected by the medium as regards direction and quantity, and

it is essentially related to a material medium. (ii) Time is required for propagation even in good conductors.

(iii) The lines of flow are either limited as in a discharge, or endless and continuous. In both cases the current depends upon two extremities, as for instance two charged conductors, or the plates of a voltaic cell.

There are thus three types of forces exerted over a distance:

Gravity, "where propagation of the force by physical lines through the intermediate space is not supposed to exist";

Radiation, "where the propagation does exist, and where the propagating line or ray, once produced, has existence independent either of its source, or termi- nation"; and

Electricity, "where the propagating process has intermediate existence, like a ray, but at the same time depends upon both extremities of the line of force." 25

Is magnetic action like any of these? Have the lines of magnetic force a physical existence, and if so, is it static like electric induction, or dynamic like

"'Ibid., P. 409. 25Ibid., p. 41 I.




an electric current? Faraday answers his three questions in the case of magnetic lines of force as follows:

(i) They have not been shown to be affected in any way by any medium other than iron. On the other hand it seems that the external lines must be curved in the case of a rectangular bar magnet in vacuum, since they begin at one pole and end at the other, and Faraday "cannot conceive curved lines of force without the conditions of a physical existence in that intermediate space. If they exist, it is not by a succession of particles, as in the case of static electric induction, but by the condition of space free from such material particles.26

(ii) No time has been shown to be required for propagation of magnetic action. (iii) The lines are dependent on opposite poles at their extremities. Magnetic

lines of force have many properties in common with those of electric induction and current, and can probably be said to be "real" in the same sense. The chief evidence for this Faraday takes to be their curvature, and the fact that current is induced in a circuit by mere motion in a magnetic field. He remains agnostic as to the precise state of matter or aether which accounts for them, whether a current or a stress or any other modification of the medium.

When Kelvin showed the mathematical equivalence of various ways of representing magnetic action and the mathematical analogy between heat flow, current flow, and electric and magnetic lines of force, Faraday declared himself strengthened in his view that the lines of force represented something physically real; in other words he was prepared to take mathematical analogy with other physical processes as evidence for physical reality. It is interesting to note, too, that he is prepared to speak of physical reality in connection with a "condition of space free from . . . material particles." In spite of the in- creasing mechanization of physics, the notion of "reality" was seldom wholly restricted to matter-in-motion.

3

Beginning with the work of Faraday and his mathematical successors, one must distinguish clearly between two ways of considering action at a distance, ways which may be called respectively the mechanical problem, and the mathematical problem.

The mechanical problem is inherited from the physics of the seventeenth century, and concerns the questions which have been considered above about the ultimate nature of matter and aether and of their manner of propagating action. All the important attempts to answer this question were in terms of a theory of substance having mechanical properties: extension, duration, motion, mass, force. In the nineteenth century, mechanical models of theories are still sought, but it is clear that in terms of these no answer to the mechanical problem of action is possible. The reason for this is that the mechanical models are

9 bid., p. 414.



348 MARY B. HESSE

no longer thought of as literal descriptions of entities existing in nature, but only as interpretations, in terms of mechanical devices, of phenomena that are described mathematically but whose ultimate nature cannot be regarded as crudely mechanical.

Take for example Maxwell's mechanical model of the electromagnetic aether. He accounts for the propagation of electromagnetic effects by a quasi-material elastic medium in which tubes of magnetic force are vortex filaments causing tension in the medium along their length and pressure laterally. The vortex motion is made possible by "idle" particles between one vortex and another; the flux of these particles in a conductor represents electric current, and their displacement in an insulating medium produces dielectric effects. This model applies to so-called free space, or aether, as well as to the interior of matter, and Maxwell showed that his equations of the electromagnetic field can be derived from it, giving the propagation of electromagnetic disturbances with the velocity of light.

If this model were intended as a description of the ultimate particles of nature, it would be difficult to say whether it involved action at a distance or not. The equations of the model are those of a continuous elastic fluid medium, but the question of whether this medium is ultimately continuous or discrete is left undecided, because it depends on more detailed knowledge of the molecular constitution of matter and aether. If matter and aether ultimately consist of discrete simple atoms, the chances are that some action at a distance, over atomic distances at least, will have to be postulated; otherwise it becomes impossible to account for the cohesion of solids. But Maxwell is concerned only to explain "action between distant bodies without assuming the existence of forces capable of acting directly at sensible distances." 27 On the question of whether aether is discrete or continuous, Maxwell remarks:

It is often asserted that the mere fact that a medium is elastic or compressible is a proof that the medium is not continuous, but is composed of separate parts having void spaces between them. But there is nothing inconsistent with experience in supposing elasticity or compressibility to be properties of every portion, however small, into which the medium can be conceived to be divided, in which case the medium would be strictly continuous. A medium, however, though ho- mogeneous and continuous as regards its density, may be rendered heterogeneous by its motion, as in Sir W. Thomson's hypothesis of vortex-molecules in a perfect liquid. The aether . . . is probably molecular, at least in this sense.28

This illustrates the sort of difficulty that arises when one attempts to take the mechanical models as literal descriptions of nature, and to wring from them an answer to the question whether action at a distance occurs in nature or not. One generally becomes involved in an infinite regress: action between bodies at finite distances is explained by continuous stress in the intervening medium; this stress is explained by the molecular constitution of the medium, which may itself involve action at a distance, and so on. There are other examples of the

"A dynamical theory of the electromagnetic field," Scientific Papers, vol. I, p. 527. 28 "Ether," ibid., vol. II, p. 774.




same sort: Helmholtz introduced terms representing viscosity into his equations of aether motion, but, as Kelvin pointed out,29 viscosity produces thermal motion of the particles themselves, and presumably viscous forces between aether particles would, if the idea is to be taken literally, involve thermal motion of particles within the particles. Again, in Lorentz's theory of the elec- tron, the force on a macroscopic charged particle is replaced by forces on its constituent electrons, and then forces on charges within electrons are spoken of.80

Such language is legitimate if it is merely a way of speaking about the appropriate mathematical equations, but no solution to the problem of action at a distance is to be found by taking such mechanical models literally. Maxwell himself is careful to explain the status of his molecular vortex model:

I propose now to examine magnetic phenomena from a mechanical point of view, and to determine what tensions in, or motions of, a medium are capable of producing the mechanical phenomena observed.31 The conception of a particle having its motion connected with that of a vortex by perfect rolling contact may appear somewhat awkward. I do not bring it forward as a mode of connexion existing in nature.32

And in a later paper:

I have on a former occasion attempted to describe a particular kind of motion and a particular kind of strain, so arranged as to account for the phenomena. In the present paper I avoid any hypothesis of this kind; and in using such words as electric momentum and electric elasticity in reference to the known phenomena

I wish merely to direct the mind of the reader to mechanical phenomena which will assist him in understanding the electrical ones. All such phrases in the present paper are to be considered as illustrative, not as explanatory.

In speaking of the Energy of the field, however, I wish to be understood literally.33

On the other hand, the mathematical aspect of the problem of action at a distance became increasingly important as the nature of mechanical models came to be better understood. The mathematical problem may be said to have been that of reinterpreting action at a distance and action by contact so that the concepts remained relevant to a physics whose fundamentals were be- coming more and more abstract and less and less mechanical, and whose structure was more easily understood in terms of mathematics than of mechanical models.

Kelvin showed in a series of mathematical papers beginning in 1842 84 that the same mathematical formalism could be used to express the laws of fluid flow, of heat flow, of electric and magnetic phenomena, and of elasticity. Thus, a source of fluid or of heat is the analogue of an electric charge, magnetic pole, or source of electric current; lines of flow are analogues of lines of force,

23Baltimore Lectures (Cambridge, 1904), pp. 9Iff.

? Lorentz, Theory of Electrons (Leipzig, 1909), pp. 13, I14-

3' "On physical lines of force," Scientific Papers, vol. I, p. 452.

3 Ibid., p. 486. 38 "A dynamical theory of the electromag-

netic field," ibid., p. 563. ` Sir W. Thomson (Lord Kelvin), Papers on Electrostatics and Magnetism, 2nd ed. (London, I884), pp. I & I5; Mathematical and Physical Papers (Cambridge, I882), vol. I, p. 76.



350 MARY B. HESSE

temperature is an analogue of potential, and so on. Kelvin also showed that Faraday's representations in terms of lines of force were consistent with the older theories of the inverse-square law, and that this followed without assuming any physical hypothesis about the nature of the lines of force. These lines are mathematically defined when the distribution of centers of force is known.

Kelvin remarks that no physical hypothesis follows from the fact of these analogies. Fourier did not deduce that heat is a material fluid from the laws of heat flow, and Coulomb did not deduce ultimate attractions and repulsions at a distance from the inverse-square law, but the analogies are bound to suggest that, if heat is propagated from particle to particle in a continuous medium, then electric and other actions may be propagated in a similar manner. Maxwell is equally cautious in his use of the analogies: he proposes to treat lines of force as if they were lines of flow of an incompressible fluid, but he adds:

The substance here treated of must not be assumed to possess any of the properties of ordinary fluids except those of freedom of motion and resistance to com- pression. It is not even a hypothetical fluid which is introduced to explain actual phenomena. It is merely a collection of imaginary properties which may be employed for establishing certain theorems in pure mathematics in a way more in- telligible to many minds and more applicable to physical problems than that in which algebraic symbols alone are used.35

General opinion in physics has, however, following Maxwell, regarded field theories derived from these analogies with fluid flow and with elastic media as the new type of continuous-action theory. Continuous action has now come to mean that each point of space can be characterized by certain mathematical quantities which describe the properties of space without implying that any mechanical events are happening there: ". . . we may regard Faraday's conception of a state of stress in the electro-magnetic field as a method of explaining action at a distance by means of the continuous transmission of force, even though we do not know how the state of stress is produced." 3 And before long physicists ceased to ask how the state of stress is produced in a mechanical sense, or even to allow that the question had any meaning.37 Faraday's criteria for continuous action began to be accepted as sufficient, namely, that the action should be affected by the medium and that its propagation should take time.

With regard to propagation in time, the work of the Continental physicists, Riemann, Neumann, Weber, Clausius, and others, on moving charges, had led to the notion of electric potential propagated from charge to charge with a finite velocity. These mathematical theories were all expressed in terms of action at a distance, and Maxwell remarks that this must be due to an a priori objection to an intervening medium:

85 "On Faraday's lines of force," Scientific Papers, vol. I, p. i6o.

TMlbid., vol. II, p. 321.

'Cf. Larmor, writing in i0oo: "It is not superfluous to repeat here that the object of a gyrostatic model of the rotational aether is not

to represent its actual structure, but to help us to realize that the scheme of mathematical rela- tions which define its activity is a legitimate conception. Matter may be and likely is a structure in the aether, but certainly aether is not a structure made of matter." (Aether and Matter, Cambridge, igoo, pp. vi fn.)



ACTION AT A DISTANCE 35I

. . . we are unable to conceive of propagation in time, except either as the flight of a material substance through space, or as the propagation of a condition of motion or stress in a medium already existing in space. In the theory of Neu- mann, the mathematical conception called Potential, which we are unable to conceive as a material substance, is supposed to be projected from one particle to another, in a manner quite independent of a medium....

But in all of these theories the question naturally occurs: If something is transmitted from one particle to another at a distance, what is its condition after it has left the one particle and before it has reached the other? 38

Maxwell's own field theory also leads to a solution representing a potential propagated from charge to charge with a finite velocity. The point at issue between the Continental school and that of Maxwell is partly a question of mathematical convenience, since formulation in terms of either action at a distance or field theory can be made to yield results that are confirmed by observations, but it is also a question of what criteria are to be used in choosing between two possible hypotheses. In addition to conformity with observation, Maxwell de- mands the possibility of thinking physically about events taking place in the medium. A mathematical formalism which is no more than an ad hoc collection of equations designed to fit the phenomena is not a sufficient aid to the imagination in trying to extend a theory. In this instance the field methods of the British school were certainly more fruitful in further developments, and this was freely admitted by later German physicists.39

The discovery that various types of physical energy are transformable one into another, and that exact numerical correspondences are found in such trans- formations, suggested that physical phenomena might be regarded as manifes- tations of a "substantial" energy in its various forms, mechanical, thermal, chemical, electromagnetic, and so on. In his Principles of Mechanics, Hertz suggested that the theory of mechanics might be rewritten in terms of space, mass and energy, instead of the traditional space, mass and force. This formulation has the advantage that energy is accessible to observation in a more direct way than force, and the use of the concept of energy does not depend upon atomic hypotheses as that of force does, for instance, in some statements of D'Alembert's Principle. Some forms of energy may be assumed to resemble a substance, although potential energy is difficult to conceive in this way, since it may be negative, and it depends on the presence of distant masses. On bal- ance, Hertz decides against this way of stating the fundamental principles of mechanics, and adopts a method which involves only space, time and mass, in which force or energy are merely derived notions.

In electromagnetism, however, the description of events in terms of the continuous flow of a quasi-substantial energy was more plausible. Poynting adopted this point of view:

A space containing electric currents may be regarded as a field where energy is transformed at certain points into the electric and magnetic kinds by means of batteries, dynamos, thermoelectric actions, and so on, while in other parts of the

88 Electricity and Magnetism (Oxford, I88I), vol. II, p. 448.

9 See for instance Helmholtz's Preface to Hertz's Die Principien der Mechanik (Leipzig, I894).



352 MARY B. HESSE

field this energy is again transformed into heat, work done by electromagnetic forces, or any form of energy yielded by currents.40

In the theories of Faraday and Maxwell, the energy of the field is not simply carried along by the currents, but resides in the intervening medium. This must follow, if continuity of motion is asserted, because, if a particle be placed in the field at a point previously empty of matter, it may acquire a kinetic energy, and this energy must come through the surrounding space. "The alternative that it appeared in the body without passing through the space immediately surrounding the body need not be discussed." 41

Poynting shows that, if the medium be regarded as the seat of energy defined at any point in terms of the electric and magnetic intensities at that point, then the energy associated with electric currents flows in towards a current-bearing wire along radii drawn from the wire, and is transformed into heat on reaching the wire itself. In a similar way, the tubes of electric intensity, which are parallel to the wire, converge on it as the current flows, and the tubes of magnetic intensity converge on it like ripples reversed in direction. The velocity of this flow depends on the dielectric and magnetic properties of the medium and on the electrical resistance of the wire. Poynting remarks that this is only a representation of Maxwell's equations, and that no observational proof of the flow is to be expected other than the evidence which already sup- ports Maxwell's theory.

4

At the end of the nineteenth century it seemed that most branches of macroscopic and molecular physics were on the way to explanation in terms of the electromagnetic field theory. Charged atomic particles were discovered, and atomic structure began to be treated by that theory. Cases of apparent action at a distance in electric and magnetic phenomena were all describable by a field theory which no longer implied a material aether, but which assigned a certain mathematical structure to space, and which described events going on in space in terms of the flow of mathematically defined concepts such as tubes of force and energy. It became indeed more correct to speak of matter as a particular modification of this field, a particular assemblage of singularities in the field, rather than to try to explain the properties of the field in terms of matter.

The only important physical phenomenon which stood outside the electromagnetic synthesis was gravitation, and this still seemed to exhibit pure action at a distance. Although the fundamental inverse-square law of gravitational force leads to the same mathematical development as Coulomb's law in electricity, several physical differences between gravitational and electric force made it difficult to regard the potential field of gravitation as more than a

40"On the transfer of energy in the electromagnetic field," Phil. Trans. Roy. Soc. (I884), P. 343.

1 Poynting, "On the connection between electric current and the electric and magnetic in- ductions in the surrounding field," ibid. (I885), p. 279.



ACTION AT A DISTANCE 353 mathematical convenience, just as the electric potentials had not modified the theory of electricity based on action at a distance before Faraday. Faraday's arguments for regarding gravity as pure action at a distance were hardly modified during the succeeding half-century. Although the mathematical analogy with electrostatics showed that one could speak of curved lines of gravitational force, it was still true that nothing in the intervening medium had been found to affect the propagation of gravity, whereas the effect of the intervening medium was one of the main reasons for asserting the physical reality of the electromagnetic field. It was still not clear whether the propagation of gravity required a finite time, and Larmor for instance wrote in I900 that its velocity of propagation was known "enormously to transcend" the velocity of light.42 Maxwell had given up the attempt to describe gravitational attraction in terms of aether action, because of insuperable difficulties about the energy of a medium in which the force between like particles is attraction, instead of re- pulsion as in the case of electric and magnetic particles.43 Until the advent of the theory of relativity, therefore, gravitation was understood no better than in the seventeenth century, when the theory of attraction was first formulated, although it had served as an essential model for the impressive and rapidly growing science of electromagnetism.

To pursue the history of the concept of action further would take us beyond the limits of classical physics, and in any case it might well be argued that with the abandonment of any attempt to describe nature literally in terms of familiar concepts such as billiard balls, elastic strings, continuous fluids and so on, the original controversy about action at a distance loses most of its point. We have perhaps gone far enough to conclude that statements containing concepts like "action," "contact," "particle," were no longer used literally in their Newtonian sense, even in the nineteenth century, but on the other hand they did not disappear from the literature of physics to be entirely replaced by mathematical formalism. Their function was to point to analogies between di- verse phenomena and to enable the new and unfamiliar to be thought about and described in terms of the familiar.

If we still desire to ask whether action in modern physics is at a distance or continuous, the only answer seems to be that it is both and it is also neither. It is both, because language and mathematical formalism derived from field theory and particle theory are still used, but it is also neither, because the language and the formalism are descriptions of natural structures which are not like the subject-matter of Newtonian mechanics, and in regard to which concepts such as "action," "particle," "distance" are only used analogically. But this is no longer the important question: a more interesting question would be, "What are the ways in which action is transmitted according to modem physics?" The answer would have to take account of the various mathematical descriptions which are used, and would indicate what new models or analogues are found to be appropriate.*

42Larmor, op. cit., p. I88.

43Scientific Papers, vol. I, p. 570.

*Note: The author wishes to acknowledge the help received from a conversation with Pro- fessor H. Dingle, and from Dr. D. McKie.



Mary B. Hesse - Acctions in Distance in Classical Physics

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