Analogies, illustrations, and working models as mechanical representations

In Station 4 Maxwell applied a variety of conceptual tools: “mechanical illustrations,”⁹¹ “working models”⁹² as well as “mental images.”⁹³ He introduced these tools to assist the mind in comprehending the (strange) phenomena of electricity and magnetism. The appeal to them was mostlydidactic.⁹⁴ Thus, for the most part they did not partake in the argumentative structure of Station 4 and therefore were not required to be consistent.⁹⁵

Maxwell developed illustrations and analogies for research purposes and then recast them as didactic tools. For example, he occasionally appealed to taut ropes and rigid rods as illustrations and analogies:

If we now proceed to investigate the mechanical state of the medium on the hypothesis that the mechanical action observed between electrified bodies is exerted through and by means of the medium, as in the familiar instances of the action of one body on another by means of the tension of a rope or the pressure of a rod, we find that the medium must be in a state of mechanical stress.

The nature of this stress is, as Faraday pointed out*, a tension along the lines of force combined with an equal pressure in all directions at right angles to these lines. The magnitude of these stresses is proportional to the energy of the electrification per unit volume, or, in other words, to the square of the resultant electromotive intensity multiplied by the specific inductive capacity of the medium.⁹⁶

[Footnote in Maxwell, 1873d, 1: 59, § 59]

* Exp. Res., series xi, 1297

According to Maxwell, electric tension

is a tension of exactly the same kind, and measured in the same way, as the tension of a rope, and the dielectric medium, which can support a certain tension and no more, may be said to have a certain strength in exactly the same sense as the rope is said to have a certain strength. Thus, for example, Thomson has found that air at the ordinary pressure and temperature can support an electric tension of 9600 grains weight per square foot before a spark passes.⁹⁷

To be sure, drawing similarities between distinct phenomena, e.g., tension in a medium and tension in a rope, helps develop an intuition for understanding the phenomenon under study, but it is not necessarily the case that consequences drawn in the familiar domain apply to the unfamiliar domain. We have sought to retain this distinction between a methodology that aids intuition and thus enhances comprehension and a methodology that advances research. We have therefore characterized analogies as weak and strong, respectively.⁹⁸

We surmise that Maxwell’s appeal to ropes, muscles, etc. in Station 4 is didactic and illustrative: no formal consequences are drawn from these analogies—these are weak analogies.⁹⁹ There was no expectation that using these analogies would help in making progress toward the goal of casting

Faraday’s verbal descriptions of his experimental results into mathematical symbolism.

There are a few strong analogies in Station 4. Consider, for example, Maxwell’s discussion of the dynamical theory of electromagnetism. The framework of the discussion is clearly mechanical: energy comes in two forms, kinetic and potential, and current—however problematic—has to be conceived kinetically:

The electric current cannot be conceived except as a kinetic phenomenon. Even Faraday, who constantly endeavoured to emancipate his mind from the influence of those suggestions which the words electric current and electric fluid are too apt to carry with them, speaks of the electric current as something progressive, and not a mere arrangement*.¹⁽¹⁰

[Footnote in Maxwell, 1873d, 2: 195, § 569] * Exp. Res., 1648

This is a most revealing remark. We noted Maxwell’s warning against the allure of analogies; and here we see him acknowledging the problem but assuming it nonetheless in order to develop a strong analogy to advance his research.

But all that we assume here is that the electric current involves motion of some kind. That which is the cause of electric currents has been called Electromotive Force. This name has long been used with great advantage, and has never led to any inconsistency in the language of science. Electromotive force is always to be understood to act on electricity only, not on the bodies in which the electricity resides. It is never to be confounded with ordinary mechanical force, which acts on bodies only, not on the electricity in them. If we ever come to know the formal relation between electricity and ordinary matter, we shall probably also know the relation between electromotive force and ordinary force.¹⁰¹

It is noteworthy that Maxwell is fully aware of the problem of inconsistency when developing such an analogy between mechanical current and electric current. He claimed that if we were to restrict the discussion to each domain, the electric and the mechanical, no confusion or incoherence would arise. And if the formal, mathematical relation between electricity and matter were to be discovered, the relation between electromotive force and ordinary force would be known. Having secured his assumptions regarding the consistency of the analogy, he proceeded:

When ordinary force acts on a body, and when the body yields to the force, the work done by the force is measured by the product of the force into the amount by which the body yields. Thus, in the case of water forced through a pipe, the work done at any section is measured by the fluid pressure at the section multiplied into the quantity of water which crosses the section.

In the same way the work done by an electromotive force is measured by the product of the electromotive force into the quantity of electricity which crosses a section of the conductor under the action of the electromotive force.

The work done by an electromotive force is of exactly the same kind as the work done by an ordinary force, and both are measured by the same standards or units.

We ... know enough about electric currents to recognise, in a system of material conductors carrying currents, a dynamical system which is the seat of energy, part of which may be kinetic and part potential.

The nature of the connexions of the parts of this system is unknown to us, but as we have dynamical methods of investigation which do not require a knowledge of the mechanism of the system, we shall apply them to this case.

We shall first examine the consequences of assuming the most general form for the function which expresses the kinetic energy of the system.¹⁰²

We have italicized two key expressions: “In the same way” and “exactly the same kind”; these expressions indicate that Maxwell developed at this juncture of Station 4 a strong analogy. Indeed, he explicitly stated that having drawn this analogy he turned to “examine the consequences,” and then proceeded to develop the dynamical theory in formal terms.¹⁰³ This approach is reminiscent of the appeal to physical analogy in Station 1. Later in this discussion of the dynamical theory, Maxwell remarked:

All this would be true, if, instead of electric currents, we had currents of an incompressible fluid running in flexible tubes. In this case the velocities of these currents would enter into the expression for T, but the coefficients would depend only on the variables x, which determine the form and position of the tubes.¹⁰⁴

Evidently, Maxwell thought that the analogy between the flow of electricity and the flow of incompressible fluid under these constraints is strong.

Maxwell’s mechanical thinking is clearly in evidence in Station 4, especially in the many weak analogies. He regarded the lines of force as tubes; for example, he called the surface generated by the motion of line of force, “a tubular surface,” and then referred to it as “a tube of induction.”^{1 5} Moreover, when Maxwell offered a “mechanical illustration of the properties of a dielectric,” his illustration came complete with “tubes” filled partly with mercury and partly with water, as well as with stopcock and piston:

To represent the case in which there is true conduction through the dielectric we must either make the piston leaky, or we must establish communication between the top of [one] tube ... and the top of [another] tube.

In this way we may construct a mechanical illustration of the properties of a dielectric of any kind, in which the two electricities are represented by two real fluids, and the electric potential is represented by fluid pressure. Charge and discharge are represented by the motion of ... [a] piston, and electromotive force by the resultant force on the piston.¹⁰⁶

Maxwell’s ingenuity is displayed here in the way he conceived the properties of a dielectric mechanically. We note the modal expression, namely, “we may construct a mechanical illustration.” Maxwell developed these weak analogies cautiously.

In Station 4 Maxwell sought “a complete dynamical theory” to account for the physics of the phenomena, not just a formal description of them. However, this methodology had to be used judiciously, and Maxwell warned the reader of possible pitfalls. Indeed, on several occasions Maxwell cautioned the reader about the misleading nature of analogies in general.¹⁰⁷

In the section, “On the induction of a current on itself,” Maxwell drew attention to Faraday’s remark that “the first thought that arises in the mind is that the electricity circulates with something like momentum or inertia in the wire.” And Maxwell added,

When we consider one particular wire only, the phenomena are exactly analogous to those of a pipe full of water flowing in a continued stream. If while the stream is flowing we suddenly close the end of the pipe, the momentum of the water produces a sudden pressure, which is much greater than that due to the head of water, and may be sufficient to burst the pipe.¹⁰⁸

On the face of it, this example expresses strong analogy: electromagnetic and hydrodynamic phenomena are exactly analogous. Indeed, the analogy of the flow of fluid in a pipe with the flow of electricity in a wire had been well impressed on the minds of many researchers. However, the analogy is in fact misleading. The inertia of the fluid in the tube depends solely on variables pertaining to what is inside the tube; it does not depend on anything outside it or on the form into which the tube may be bent. Maxwell contrasted the physical analysis of the fluid flowing in a pipe to the wire conveying a current and stated that in the latter case variables external to the tube/wire are critical to the flow phenomena. He then added an insightful remark:

It is difficult, however, for the mind which has once recognised the analogy between the phenomena of self-induction and those of the motion of material bodies, to abandon altogether the help of this analogy, or to admit that it is entirely superficial and misleading. The fundamental dynamical idea of matter, as capable by its motion of becoming the recipient of momentum and of energy, is so interwoven with our forms of thought that, whenever we catch a glimpse of it in any part of nature, we feel that a path is before us leading, sooner or later, to the complete understanding of the subject.¹⁰⁹

Maxwell cautioned the physics community against such misleading conceptions that come from uncritical habits of thinking. This was probably one reason for his increasing use of weak analogies in Station 4. Previously, analogy was part of the argument; now it is principally didactic and illustrative.

Here the focus is on the misleading analogy between electric current and the current of a fluid:

It appears to me, however, that while we derive great advantage from the recognition of the many analogies between the electric current and a current of a material fluid, we must carefully avoid making any assumption not warranted by experimental evidence, and that there is, as yet, no experimental evidence to shew whether the electric current is really a current of a material substance, or a double current, or whether its velocity is great or small as measured in feet per second.¹¹⁰

Still, this was a beginning:

A knowledge of these things would amount to at least the beginnings of a complete dynamical theory of electricity, in which we should regard electrical action, not, as in this treatise, as a phenomenon due to an unknown cause, subject only to the general laws of dynamics, but as the result of known motions of known portions of matter, in which not only the total effects and final results, but the whole intermediate mechanism and details of the motion, are taken as the objects of study.¹”

Maxwell entertained the possibility that electricity might, after all. be a current of some kind of “material substance.” Moreover, he envisaged a complete theory of electrodynamics that would cover comprehensively the entire chain from cause to effect, that is, the intermediate mechanisms and the known motions of the known portions of matter, as well as the governing laws. But meanwhile he was not committed to any mechanism at the micro-level.

This is the epistemic context in which Maxwell commented on his “working model,” as he now, in Station 4, called his “mechanical hypothesis” of Station 2.¹¹² In a note (§ 831) Maxwell quoted at length Thomson’s discussion of Faraday’s discovery of the magnetic influence on

Station 4 (1873) 181 light.¹¹³ As Thomson remarked, “the magnetic influence on light discovered by Faraday depends on the direction of motion of moving particles.”¹¹⁴ And Thomson continued:

I think it is not only impossible to conceive any other than this dynamical explanation ... but 1 believe it can be demonstrated that no other explanation of that fact is possible.¹¹³

We have omitted the physical content regarding the direction of motion of the particles in order to emphasize the grammatical structure of this claim: the explanation stands in a one-to-one relation to the phenomenon—so certain was Thomson of his proposal. It appeared to Thomson that

Faraday’s optical discovery affords a demonstration of the reality of Ampere’s explanation of the ultimate nature of magnetism; and gives a definition of magnetization in the dynamical theory of heat ... Whether this matter is or is not electricity, whether it is a continuous fluid interpermeating the spaces between molecular nuclei, or is it itself molecularly grouped; or whether all matter is continuous, and molecular heterogeneousness consists in finite vortical or other relative motions of contiguous parts of a body; it is impossible to decide, and perhaps in vain to speculate, in the present state of science.¹¹⁶

For Thomson, then, the exact mechanism cannot be determined—the hypothesis is unproven; but the theory holds. This argument impressed Maxwell who quoted it at length. At this juncture Thomson referred to the hypothesis of molecular vortices which Rankine put forward and to which both Thomson and Maxwell appealed in their explanations of various phenomena of magnetism. At the end of the quotation of Thomson’s text Maxwell referred to his own work on molecular vortices.¹¹⁷

Maxwell ended the quotation, but Thomson continued, “I append the solution of a dynamical problem for the sake of the illustration it suggests for the two kinds of effect on the plane of polarization ..He then described a mechanical arrangement that consists of a cord attached to the two ends of a horizontal arm which is made to rotate around a vertical axis at its midpoint; a second cord is tied at the midpoint of the first cord, bearing a weight. The two cords are assumed to be “perfectly light and flexible” and the problem is to determine the motion of this artificial device “when [it is] infinitely little disturbed from its position of equilibrium.”¹¹⁸ According to Thomson, the various solutions offer illustrations for the possible mechanism of magnetism:

From these illustrations it is easy to see in an infinite variety of ways how to make structures, homogeneous when considered on a large enough scale, which (1) with certain rotatory motions of componentparts having, in portions large enough to be sensibly homogeneous, resultant axes of momenta arranged like lines of magnetic force, shall have the dynamical property by which the optical phenomena of transparent bodies in the magnetic field are explained', (2) with spiral arrangements of components parts, having axes all ranged parallel to a fixed line, shall have the axial rotatory property corresponding to that of quartz crystal', and (3) with spiral arrangements of component groups, having axes totally unarranged, shall have the isotropic rotatory property possessed by solutions of sugar and tartaric acid, by oil of turpentine, and many other liquids) ^w

Thomson was convinced that magnetism is to be explained mechanically; he realized, however, that there are infinite modes of mechanical explanations.

This is the background to Maxwell’s opening remark in § 831 of Station 4: “the whole of this chapter may be regarded as an expansion of the exceedingly important remark of Sir William Thomson in the Proceedings of the Royal Society, June 1856.” We consider the remark that follows a key to understanding Maxwell’s view of working models:

I think we have good evidence for the opinion that some phenomenon of rotation is going on in the magnetic field, that this rotation is performed by a great number of very small portions of matter, each rotating on its own axis, this axis being parallel to the direction of the magnetic force, and that the rotations of these different vortices are made to depend on one another by means of some kind of mechanism connecting them.¹²⁰

This is the celebrated mechanical conception of molecular vortices which Maxwell introduced in Station 2 and to which Maxwell explicitly referred at this juncture of Station 4.¹²¹ Curiously, in Station 2 Maxwell referred to this mechanism as a hypothesis, whereas here, in Station 4, he called it a “working model”;

The attempt which I then made to imagine a working model of this mechanism must be taken for no more than it really is, a demonstration that mechanism may be imagined capable of producing a connexion mechanically equivalent to the actual connexion of the parts of the electromagnetic field. The problem of determining the mechanism required to establish a given species of connexion between the motions of the parts of a system always admits of an infinite number of solutions. Of these, some may be more clumsy or more complex than others, but all must satisfy the conditions of mechanism in general.¹²²

This usage of “working model” is most idiosyncratic. At the time of Maxwell, “working model” was primarily related to submissions to the

Station 4 (1873) 183 patent office and the like (a miniature machine which is able to perform the same functions as the full-scale machine it is intended to represent).¹²³ The expression “working model” in the sense of “working hypothesis” does not seem to be attested before 1873. The context makes it clear that Maxwell did not consider the scheme of molecular vortices to be a false theory: the scheme is just unproven and one of many (mechanical) possibilities for assisting in understanding the phenomenon, namely, the relation between light and magnetism.¹²⁴

Maxwell commented on the limitations of what he called his “working model,” namely, a “mechanism may be imagined,” which is capable of accounting for the phenomena. But at the same time, he recognized, echoing the view of Thomson, that an “infinite number of solutions [are possible].” Still, once the theory is complete and confirmed, its scaffolding —be it a “model” or a “mechanical illustration”—becomes superfluous.¹²⁵

Maxwell’s remark is both autobiographical and philosophical. In the first place he recalled his attempt in Station 2 “to imagine a working model,” which at the time he called a “mechanical illustration,” composed of ordinary materials, based on his molecular-vortex hypothesis.¹²⁶ So Maxwell took account of his earlier work. But then he added a philosophical note of great importance, arguing that his theory does not depend on any specific mechanism. Nevertheless, he seemed to adhere to the view that there is an underlying mechanism, involving vortices and stress, which linked electromagnetic phenomena. For him, however, it was an open question whether the mechanism underlying these phenomena could be discovered and experimentally verified. Despite the absence of a clear statement to this effect, it is likely that Maxwell did not believe that such a mechanism could be found. As was the case with Thomson, the exact mechanism cannot be determined; the hypothesis is unproven, but the theory holds.