Foundations of Physical Theory: Function of Models

A Guest Post from George Francis FitzGerald (1851-1901)

Electrical and magnetic attractions and repulsions are familiar to most people. They know that amber and sealing-wax when rubbed attract small pieces of paper, and that the Earth causes magnetic needles to point more or less north and south. The first notion that struck investigators of these facts was that the amber and the Earth "possessed the property of attracting" certain objects at a distance from them. This certainly expressed all the facts as then known and expressed them simply. Such actions at a distance were easily expressed in mathematical symbols, and their laws were thus investigated and the whole language of the sciences was framed so as to fit in with this view of the phenomena.

That these actions can be expressed in terms of symbols that presupposed an action at a distance does not, however, in any way prove that the action is not by means of some intervening medium. Take for instance a simple mechanical illustration. Suppose that you pull a body along by means of a string.

A child pulls a toy with a string.

This action may be expressed by symbols that only represent that the distance between you and the body is invariable, and that do not in any way involve the existence of a string, or indeed of any medium that transmits the action. It is only when you wish to take account of actions that take some time to be transferred from you to the body, it is only then that it is necessary mathematically to introduce symbols expressly referring to a medium.

We may gather from this how it was absolutely necessary in the case of the action of light to introduce the notion of something existing between the Sun, for instance, and the Earth, while it was not necessary, as far as was known, to make a similar assumption in regard of electrical actions. In the case of light it was known that it took time to be transmitted, while in the case of electrical actions they were, as far as was known, instantaneous, and could consequently be easily represented as taking place by an action at a distance. Although there are no experiments that prove certainly that either electric or magnetic actions are not instantaneous, nevertheless it is well worth while investigating how they may be due to a medium, for if they can be shown to be explicable by a medium possessing the same properties as are required in order to explain the transmission of light, the hypothesis that they are due to a medium will at once assume great importance, as these actions will then be explained as due to a known cause.

George Francis FitzGerald (1851-1901)

Before describing, in general terms, the sort of way in which it has been shown that electric and magnetic phenomena may be due to a medium, it may be well to call attention to how very little we know of the constitution of the media we use every day for transmitting force. Take the case of a beam supporting a weight. Nobody knows what the structure of its molecules is, by means of which they are able to resist compressive and shearing stress. But nobody on this account doubts the usefulness of considering and calculating these stresses.

Simple supported beam with uniform load.

Similarly, although nobody has been able to frame any satisfactory hypotheses as to the constitution of the medium that causes electric and magnetic and luminiferous actions, nevertheless it is obviously very important to be able to calculate the stresses that it must be subject to at its various points in order that it may cause these actions. In the case of a clock or watch spring nobody knows what is the exact structure of its molecules by means of which it is able to act as a reservoir of energy, which we put into it when we wind up the clock or watch, and which is gradually expended in keeping the works going.

Pocket watch.

The fact that we do not know this does not in the least diminish from the importance of our knowing that there is a spring, and that the action is not a pure action at a distance between the axle that we turn with the key and the barrel that turns the wheels. Similarly it is of importance to know where in the medium the energy of an electrified system is stored, although we may not be able to state what the exact structure of the medium must be in order that it may be capable of acting' as a reservoir of energy.

The calculation of electric and magnetic actions is a subject upon which mathematicians have expended some of their highest abilities, and the simplicity of the general laws of these actions lends itself to mathematical calculation to no ordinary degree. Mathematically it was simpler to represent the actions as actions at a distance, and not to introduce symbols representing the existence of a medium intervening between the bodies concerned. Hence it was that the whole of the language of the science was framed upon this hypothesis, and it seems likely that the counter-hypothesis of action through a medium might have had to wait as long as the undulatory theory of optics did before it received due consideration, only for the investigations of one who never received a mathematical training.

To one of less ability than Faraday this would have placed him at an immense disadvantage in comparison with trained investigators. In his case, however, his wonderful power of reasoning from things without the use of symbols not only placed him in the forefront of scientific investigators, but enabled him to see into the method of working of the phenomena in a way that escaped mathematicians who reasoned by the use of symbols. To Faraday is due the credit of having seriously called attention to the notion that electric and magnetic actions were due to a medium pervading the space between the acting bodies. This hypothesis of Faraday's was expressed in mathematical symbols by Clerk Maxwell, who then made the great discovery that a medium which possessed the properties required in order to explain electric and magnetic actions would transmit vibrations like light.

Michael Faraday (1791-1867), left, and James Clerk Maxwell (1831-1879), right.

There is a term describing the condition of the medium that it is necessary to explain by examples which, although they may not be like the condition of the medium, are so far analogous to it, that we may in many respects reason safely from one to the other. This term is "polarization." A thing is said to be polarized when it possesses different properties in different directions. Thus a magnet is said to be polarized because it possesses different properties at its ends. A ray of light is said to be polarized when it possesses different properties at different sides. A mechanical illustration may be imagined by considering two wheels, on parallel axes, connected by an India rubber band passing over both. As long as the two turn equal amounts, the two connecting parts of the band will be equally tight If, however, one wheel turn more than the other, two parts of the band will no longer be equally tight, and the system may be said to be polarized, for its two sides are no longer the same, for one is tight and the other loose.

Oliver Lodge, Modern Views of Electricity, 3rd ed. 1907, p. 254.

The direction of the polarization would, in this case, be the line joining the tight and loose parts of the band, and at right angles to the line joining the centres of the wheels. In this case the polarized system is a reservoir of energy, and there are forces tending to cause rotation of the wheels. It is of great importance to understand what sort of thing is meant by "polarization," because the whole of Faraday's theory is founded on supposing the medium to consist of elements which are capable of being polarized, and it is supposed that a certain amount of energy is stored in the polarized elements. [See FitzGerald, “On a Model Illustrating Some Properties of the Ether,” Scientific Proceedings of the Royal Dublin Society, January 1885, pp. 407-419].

Without introducing considerable complications, it is not very easy to explain how such a supposition as this can explain electric and magnetic forces ; but some general considerations will show how such a supposition can be made to explain the existence of some forces between bodies. Consider the simple case of two bodies in a medium in which energy in any form is stored, and suppose that the amount of this energy depends on the distance between the bodies in such a way that the farther the bodies are apart the more energy there will be in the medium. Under these circumstances there must be some forces tending to draw the bodies together ; for, by hypothesis, in separating them, work must be done in increasing the energy in the medium, and as work is done in separating them there must be a force drawing them together. Now there are two ways in which energy may exist in such an element as has been described, consisting of two wheels connected by an elastic band. It may exist as stored in the elastic band when the element is polarized, or it may exist as energy of rotation of the wheels. Now Clerk Maxwell's theory is that electric actions are due to polarization of the elements of the medium, and that magnetic actions are due to its elements being reservoirs of kinetic energy. It is, of course, quite possible that all energy is really kinetic, but until some workable hypothesis as to the structure of the medium has been discovered that will explain its property of storing energy by being polarized, and which explains this property by storing kinetic energy, until some such hypothesis has been discovered it is well to distinguish the energy that is stored by the polarization of the elements of the medium from what is stored in the form of motion within those elements.

Before proceeding to explain how the electric and magnetic conditions of the ether may be illustrated by a model, it may be well to say a few words upon the general question of physical analogues. There is some danger in using physical analogues as assistants to our reasoning. There is no doubt but that a concrete mechanism that we can distinctly picture the working of is enormously easier to reason about than one of whose structure we know nothing, but only know general laws of its action. The danger is that we may be satisfied with an analogy, and mistake it for a likeness.

For instance, the laws of projectiles and those of light are in a great many respects the same, so that an analogy between the propagation of light and of a projectile can be drawn, and it was drawn and was supposed to be a likeness for more than a hundred years. From the time of Newton until far into the present century, light was supposed by a large number of scientific men to be due to projectiles emitted by luminous bodies. All reasoning about nature is, however, in part necessarily reasoning from analogy. Whenever we express a general law about things whose intimate structure we do not know, our language may be, for aught we know, at least partly of the nature of analogy. When we describe a current of electricity as running, we are almost certainly using a mere analogy. It is usual to speak of chemical bonds; and many people think of light vibrations as simple to-and-fro motions of the elements of an elastic jelly : in both these cases we ought to be careful to recollect that " bond " and “vibration" are capable of a much wider meaning than their usual simple ones, and that the wider meaning is possible, because some of the laws of a great variety of different things are the same. As a very simple example of this, the laws of the addition of money, length, weight, etc., are all the same : 12 pence and 3 pence is 15 pence, and 12 yards and 3 yards is 15 yards. In a similar way the laws of conduction of heat and of diffusion of gases are in many respects the same.

Notwithstanding the danger of our mistaking analogies for likenesses, there is a great advantage in studying analogues.

It has been objected to the quaternion analysis that it does not analyse enough, but leaves the quantities we deal with nearly as complicated as to their laws as the phenomena they represent. This is certainly in some respects a disadvantage. It might similarly be objected to a mechanical model of the ether that it was as complicated as the ether itself, or possibly much more so, and could consequently be of but little use in disentangling its laws. In both cases the same answer is possible.

By analysing too much we lose a comprehensive view of the subject. Our minds are overwhelmed with the details, and we fail to grasp the action as a whole. In the case of a mechanical model of the ether, we have before us a structure which we may easily conceive, and with the method of whose working we are familiar ; and so we can reason about it and discover what it should do without being troubled at every turn to realise our analysis, for the realisation is so easy and familiar that it gives our minds no trouble.

As an example, it is worth observing that if the forms of energy were as familiar a conception as eggs and money, people would find it as easy to reason about its transformation as they are about the number of eggs the old woman brought to the market and sold at one dozen at 3 a penny and so forth. It is because at every turn people lose the thread of the argument by the difficulty of realising what they are dealing with that they find it difficult at first to reason consistently about a new subject.

It is on account of this that it is worth while studying analogies between things with which we are not familiar and those with which we are. It is for a similar reason that having made ourselves familiar with the laws of algebra we can so readily apply it to geometrical analysis. A model of the ether is like a complicated algebraic formula that can be interpreted in terms of a known analogy so as to represent the laws of a surface ; it is not like the surface, but we may deduce the laws of one from the other if we attend to the laws of the analogy. We must not press analogies too far. To suppose that the ether is at all like the model I am about to describe would be as bad a mistake as to suppose a sphere at all like x² + y² + z² = r², and to think that it must in consequence be made of paper and ink…..

Afterward:

The refusal on principle to attempt to understand reality is in large part what is wrong with contemporary physics. The contrast with the thinking of a genuine innovator like FitzGerald is striking.

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