This will require three separate passages. All three excerpts come from Basil Mahon’s The Man Who Changed Everything: The Life of James Clerk Maxwell. The first excerpt concerns Maxwell’s fluid analogy and comes from Chapter 5: “Blue and Yellow Make Pink, Cambridge 1854 – 1856,” pages 59 – 65.

Our second excerpt explains Maxwell’s vortex theory and comes from Chapter 7: “Spinning Cells, London 1860 – 1862”, pages 95 – 110:Everyone marveled at [Michael Faraday’s] experimental genius, but few placed equal value on his ability as a theoretician.

Maxwell was one who did. Reading through Faraday’s Experimental Researches, James was struck at first by Faraday’s openness and integrity:

Reading further, James came more and more to recognize the power of Faraday’s thinking. The great man’s modest description of his task as ‘working, finishing, publishing’ gave no idea of the intellectual strength and subtlety which pervaded his work. Discoveries had not come easily: great leaps of imaginative thought had been needed, but these were always subjected to the most searching interrogation before Faraday would commit himself to an opinion. James could see that Faraday’s notion of lines of force was far from a vague flight of fancy; it was a serious theory.Faraday … shows us his unsuccessful as well as his successful experiments, and his crude ideas as well as his developed ones, and the reader, however inferior to him in inductive power, feels sympathy even more than admiration, and is tempted to believe that, if he had the opportunity, he too would be a discoverer.

Faraday had puzzled hard and long about the way magnets behave. He concluded that a magnet is not just a piece of iron with interesting properties but the center of a system of invisible curved tentacles, or lines of force, which spread through all space. In Faraday’s scheme, magnetic attraction between a north pole and a south pole arises because every tentacle runs from one pole to the other and exerts a pull along its length like a piece of elastic. On the other hand, magnetic repulsion, as between two north poles, comes about because the tentacles push sideways against each other like pieces of compressed rubber. Near the magnet’s poles the tentacles are packed closely together and the forces are great; far away the tentacles are sparse and the forces small. One can see why Sir George Airy called the idea ‘vague and varying’. But anyone who has seen the beautiful curved patterns made by iron filings sprinkled on a sheet of paper near a magnet can see exactly what Faraday had in mind. The filings become little magnets and are pulled into perfect alignment along the lines of force, each line of filings being kept separate from neighboring lines by the sideways repulsion.

To explain electrostatic forces between electrically charged bodies, Faraday postulated electric lines of force which behaved in a similar way. And he went further by suggesting that a magnet induces a kind of tension in any nearby loop of metal wire, which will make a current flow if either magnet or wire is moved. He called this the electrotonic state.

To James, Faraday’s ideas rang strong and true; all they lacked was numerical expression. If he could find a way to represent lines of force mathematically, he should be able to show that all the known formulae for static electrical and magnetic effects could be derived from them just as well as from the hypothesis of action at a distance. This would confound Faraday’s critics and, although it would not itself be a new theory, it might give James a base from which he could develop one.

The present task was to deal with stationary lines of force. This was a limited objective: a complete theory of electricity and magnetism would need to deal also with moving lines, but that could come later. The static problem was essentially geometrical, but it was a more complex geometry than that of Euclid. The curving lines of force filled all space and at every point a force acted with a particular strength along the line that passed through it. This was the geometry of vectors, quantities which vary continuously through space in both magnitude and direction. Although the formal mathematics of vectors was in its infancy some good work had already been done by others and James was able to make use of it. We shall come to that, but first James wanted to get as far as he could by simply following the steps in Faraday’s reasoning, giving the lines of force a rigorous quantitative interpretation but using as few mathematical symbols as possible.

But where could one start? Were there any other physical phenomena which behaved something like lines of force and were mathematically amenable? If so, perhaps an approach by analogy might work. Inspiration was at hand. It came from William Thomson, the Glasgow University colleague of cousin Jemima’s husband Hugh Blackburn, whom James had first met while still at school. Only 7 years older than James, Thomson was already an influential figure and well established as Professor of Natural Philosophy at Glasgow University. While still an undergraduate at Cambridge he had made a remarkable discovery – the equations for the strength and direction of electrostatic force had the same mathematical form as those describing the rate and direction of a steady flow of heat through a solid material. It seemed a bizarre association, static force and moving heat, but James saw a chink of light; surely this was an idea worth pursuing. He knew Thomson well by now and wrote announcing his wish to ‘poach on your electrical preserves’. Thomson enjoyed the role of mentor and was delighted to agree. Like James, he was more concerned with advancing human knowledge than with building his own reputation, and in any case was by now deeply engaged in other work, including the most exciting technology of the day – the telegraph.

William Thomson’s comparison of electrostatic force to heat flow was an inspired insight but did not quite fill the bill. James wanted an analogy that the mind could grasp more readily than the flow of heat. He was also keen, at this early stage, to avoid even the appearance of proposing any particular physical mechanism for lines of force. The analogy he chose was that of an imaginary weightless and incompressible fluid flowing through a porous medium: the streamlines of flow would represent lines of either electric or magnetic force and he could vary the porosity of the medium from place to place to account for the particular electrical or magnetic characteristics of different materials.

Faraday had thought of lines of force as discrete tentacles. James merged them into one continuous essence of tentacle called ‘flux’ – the higher the density of flux at a point, the stronger the electrical or magnetic force there. The direction of fluid flow at any point corresponded to the direction of the flux and the speed of flow to the flux density. As a device to track the motion of the whole body of swirling fluid, James constructed imaginary tubes for it to flow along. The fluid behaved as though the tubes had real walls, because the lines of flow never crossed one another and the whole system of tubes fitted together, leaving no gaps. The fluid traveled fast where the tubes were narrow and slower where they widened out. Electric and magnetic flux, although stationary, were similarly contained in tubes: and, by analogy, forces were strong where the tubes were narrow and the flux dense, and weak where the tubes were wide and the flux sparse.

What made the fluid move was pressure difference. Along each tube the fluid flowed from high pressure to low and the flow at any point was proportional to the pressure gradient there. Correspondingly, flux was caused by differences in electrical potential (voltage) or magnetic potential and the flux density at any point was proportional to the potential gradient, which he called the intensity of the field. Points of equal pressure or potential lay on imaginary surfaces which crossed the tubes everywhere at right angles, fitting together like a continuous set of onion skins.

James elaborated the analogy to account for every characteristic of static electricity and magnetism. Positive and negative electrical charges were represented by sources and sinks of fluid, and materials with different susceptibilities to electricity and magnetism by media with different porosity. Electrical conductors were represented by regions of uniform pressure where the fluid was still. And by a separate analogy he found that the flow of the same fluid could also represent an electric current flowing through a conducting material.

It all came together. The fluid analogy not only gave formulae identical to those from the action at a distance hypothesis, it accounted for some electric and magnetic effects which occurred at the boundary between different materials but could not be explained by action at a distance.

The key to it all was that the fluid was incompressible: every cubic centimeter of space always contained the same amount of fluid, no matter how fast the fluid was moving. This automatically led, for example, to the result that electric and magnetic forces between bodies varied inversely as the square of their distance apart – it was simply a matter of geometry.

James had vindicated Faraday and turned his ‘vague and varying’ lines of force into a new and mathematically impeccable concept, the field. So far, he had only dealt separately with static electric and magnetic fields. The difficult problem of working out how the fields interacted when they changed lay ahead, but there was more he could do, even now. It concerned the well-known effect that a moving magnet makes a current flow in any nearby loop of wire. As we have seen, Faraday thought this happened because the magnet, even when stationary, induced a mysterious ‘electrotonic state’ in the wire, which caused a current as soon as the magnet (or the wire) moved. Odd though it seemed, James felt in his bones that the idea was sound and he set out to find a way to express the electrotonic state mathematically.

For this he had to deviate from his plan to use only pictorial descriptions and simple equations. He needed some power tools and looked into what had already been done in the mathematics of vector quantities. It had been mainly the work of three men. The first was George Green, a Nottinghamshire miller who had taught himself mathematics and eventually managed to get to Cambridge University at the age of 40 but died in 1841, 2 years after gaining a fellowship. The second was none other than William Thomson, who in 1845 found a forgotten paper which Green had written 17 years earlier and saw at once that it was pure gold. He had the paper re-published and started to develop Green’s ideas. The third was George Gabriel Stokes, Lucasian Professor of Mathematics at Cambridge, who was a good friend to Thomson and, later, to Maxwell and a valued source of advice and encouragement to both.

Making good use of their work, James brought legitimate status to Faraday’s electrotonic state by deriving a mathematical expression for it. As usual, it was his intuition that pointed the way. There were two established laws connecting electricity and magnetism. One gave the total magnetic force round an imaginary loop surrounding a wire carrying a given current. The other said how much electric current would be generated in a loop of wire when the magnetic flux passing through the loop increased or decreased at a given rate. These laws appeared to be sound but James felt that they did not quite get to the nub of the matter, because they dealt only with accumulated quantities summed round or through the loop. The only way to get a proper understanding, he thought, would be to look at the intimate relationship between electricity and magnetism in a single small part of space, so he set about re-expressing the laws in what mathematicians call differential form – using vectors at a single point rather than accumulated quantities summed through or round the loop. When he did this he found that one of the vectors exactly fitted Faraday’s concept of the electrotonic state: it had no effect when at a uniform and steady value but when it varied over time or space it gave rise to electric or magnetic forces. A success, but to James’ mind only a partial one – he had given a mathematical definition but could not think how to interpret the symbols physically, even in terms of an analogy. So the electrotonic state still held on to some of its mystery – for the present.

This was as far as he could go for now. From Faraday’s notion of lines of force he had created the field, a concept that became the standard pattern for twentieth century physics but was startlingly new to scientists of the time, who were used to thinking of reality in terms of material bodies with passive empty space in between. His system of interlocking forces, fluxes and potentials is exactly what physicists and electrical engineers use today. Even so, he had added little to Faraday’s ideas beyond giving them mathematical expression, and the resulting equations, although satisfactory, accounted only for static electrical and magnetic effects. He still had to confront the task of extending the work to include changing fields. These appeared to interact in ways quite unlike any other physical phenomenon, and he must have felt he was facing a sheer rock-face. But the field concept gave him a solid base, and by expressing the electrotonic state mathematically he had at least made a toe-hold on the climb.

In his Cambridge paper On Faraday’s Lines of Force he had found a way of representing the lines of force mathematically as continuous fields, and had made a start towards forming a set of equations governing the way electrical and magnetic fields interact with one another. This was unfinished business and he now felt ready to make a serious attempt to settle it. He had made progress so far by using the analogy of a swirling body of incompressible fluid – the pressure in the fluid corresponding to electric or magnetic potential and the direction and speed of flow represented the direction and strength of either an electric or a magnetic field. By extending the imagery, so that, for example, sources and sinks of fluid represented electrically charged surfaces, he had been able to derive all the important formulae for static electricity and magnetism. He had also managed to bring steady electric currents and their effects into the scheme by using the fluid analogy in a different role to represent the flow of electricity.

The analogy had served well but could take him no further because it worked only when electrical and magnetic fields were static and electric currents steady. As soon as anything changed, the fields acted in a way that was nothing like the smooth flow of a fluid; in fact their behavior was completely different from that of any known physical process. So to go further he had to find a new approach.

Two courses seemed to be open. One was to desert Faraday and fields, and assume that all effects result from action at a distance between magnetic poles and electrical charges or currents. This was the approach taken by Simeon-Denis Poisson and Andre-Marie Ampere, who had derived the original formulae for static fields and steady currents which James had re-derived by the field approach. It was also the basis of an attempt at a complete theory by Wilhelm Weber, which was mathematically elegant and offered an explanation for most of the known effects. But Weber had made a critical assumption – that the force between two electrical charges depends not only on their distance apart but also on their relative velocity and acceleration along the straight line joining them. James respected Weber’s work but his intuition bridled at this assumption and, more generally, at the whole action at a distance concept.

He therefore chose the second route, which was to go beyond geometrical analogy and make an imaginary mechanical model of the combined electromagnetic field – a mechanism that would behave like the real field. If he could devise a suitable model, the equations governing its operation would also apply to the real field.

As we have seen, all the known experimental results in electricity and magnetism could be attributed to four types of effect; to gain the day, James’ model would have to account for all of them:

1. Forces between electrical charges: unlike charges attract; like charges repel, both with a force inversely proportional to the square of the distance between the charges.

2. Forces between magnetic poles: unlike poles attract; like poles repel, both with a force inversely proportional to the square of the distance between them; poles always occur in north/south pairs.

3. A current in a wire creates a circular magnetic field around the wire, its direction depending on that of the current.

4. A changing total magnetic field, or flux, through a loop of wire induces a current in the wire, its direction depending on whether the flux is increasing or decreasing.

And it would need to do so precisely, so that all the established formulae involving electric charge and current, magnetic pole strength, distances and so on could be derived from the model, together with any new formulae.

James began with effect 2, magnetic forces. For his model he needed to envisage a medium filling all space which would account for magnetic attraction and repulsion. To do this, it would need to develop tension along magnetic lines of force and exert pressure at right angles to them – the stronger the field, the greater the tension and the pressure. And to serve its purpose as a model the imaginary medium would have to be built from components which bore some resemblance to everyday objects.

It seemed an impossible task, but James’ idea was amazingly simple. Suppose all space were filled by tiny close-packed spherical cells of very low but finite density, and that these cells could rotate. When a cell rotated, centrifugal force would make it tend to expand around the middle and contract along the spin axis, just as the earth’s rotation causes it to expand at the equator and flatten at the poles. Each spinning cell would try to expand around the middle but its neighbors would press back, resisting the expansion. If all the cells in a neighborhood spun in the same direction, each would push outwards against the others; they would collectively exert a pressure at right angles to their axes of spin.

Along the axes of spin the opposite would happen. The cells would be trying to contract in this direction and there would be a tension. So if the spin axes were aligned along lines in space, these lines would behave like Faraday’s lines of force, exerting an attraction along their length and a repulsion sideways. The faster the cells spun, the greater would be the attractive force along the lines and the repulsive force at right angles to them – in other words, the stronger the magnetic field.

So the field would act along the spin axes of the cells. But which way? Magnetic force is conventionally defined as acting from north to south pole. James built an extra convention into his scheme: the sense of the field would depend on which way the cells were spinning – it would be in the direction a right-handed screw would move if it rotated the same way; if the cells reversed their spin, the field would reverse too.

But if the cells occupied all space, why were they not apparent? And how could they exist in the same space as ordinary matter? James was not put off by such awkward questions. It was, after all, only a model. The cells’ mass density could be so low as to offer no perceptible obstruction to ordinary matter. As long as they had some mass and rotated fast enough they would generate the necessary forces.

The scheme did not yet explain how different materials could have different magnetic characteristics. For example, iron and nickel had a high magnetic susceptibility – they could be readily magnetized – whereas other substances, like wood, seemed to be even less receptive to magnetism than empty space. James solved this problem with his customary sureness of touch. Where cells occupied the same space as an ordinary substance their behavior would be modified according to the magnetic susceptibility of the substance. The modification was equivalent, in mechanical terms, to a change in the mass density of the cells. In iron, for example, they would become much more dense than in air or empty space, thereby increasing the centrifugal forces, and hence the magnetic flux density, for a given rate of spin.

Here was the basis of a model. The spin axes of the cells gave the direction of the magnetic field at any point in space: their density and rate of spin determined its strength, and the model provided exactly the right equations for effect 2, magnetic forces in static situations.

So far so good. But there were two problems. First, what set the cells in motion? And second, the cells along one line would be spinning in the same direction as those in neighboring lines, so that where two surfaces made contact they would be moving in opposite directions, rubbing awkwardly against one another. Amazingly, James solved both problems with a single stroke.

To avoid the cells rubbing against one another, he tried putting even smaller spherical particles between the cells. They would act like ball bearings, or like the ‘idle wheels’ engineers put between two gear wheels which need to rotate in the same direction. The idea seemed crazy but James persevered and suddenly things began to fall into place. Suppose the little idle wheels were particles of electricity. In the presence of an electromotive force they would tend to move along the channels between the cells, constituting an electric current, and it would be this movement that set the cells spinning.

But everyone knew that currents could flow only in substances which were conductors, like metals. In insulators like glass or mica, or in empty space, there could be no currents. So James proposed a second way in which the behavior of the cells would be modified according to the type of substance which shared their space. In an insulator the cells, or perhaps local groups of cells, would hold on to their little particles so that they could rotate but not move bodily. But in a good conductor like a copper wire the particles could move bodily with very little restriction and a current would flow. In general, the lower the electrical resistance of the substance, the more freely the particles could move.

An essential feature of James’ little particles was that they had rolling contact with the cells – there was no sliding. Where the magnetic field was uniform the particles would just rotate, along with the cells. But if the particles in a conductor moved bodily without rotating, they would cause the cells on either side of the current to spin in opposite directions, exactly the condition to create a circular magnetic field around a current-carrying wire – effect 3. If the particles rotated and moved, the circular magnetic field due to their movement would be superimposed on the linear one due to their rotation.

So, by this extraordinary assemblage of tiny spinning cells interspersed with even smaller ‘idle wheel’ particles, James had succeeded in explaining two of the four main properties of electricity and magnetism. A highly satisfactory start, but there was much more to do. The next task was to explain effect 4: a changing magnetic flux through a loop of wire induces a current in the wire. James chose to explain an equivalent effect – that when a current is switched on in one circuit, it induces a pulse of current in a nearby but separate circuit by creating a magnetic field that links the two. He drew a diagram to illustrate his argument, giving the spherical cells a hexagonal cross-section ‘purely for artistic reasons’. We can see it, slightly adapted for our purpose, in Figures 2a-d.

The diagram shows a cross-section of a small region of space. The idle wheel particles along AB are in a wire which is part of a circuit with a battery and a switch, initially open. Those along PQ are in another wire which is part of a separate circuit having no battery or switch. The idle wheels along AB and PQ are free to move because they are in conductors, but others in the neighborhood are in a non-conducting material and can only rotate in their fixed positions. AB and PQ are, of course, impossibly thin wires and impossibly close together, but this is to just keep the diagram compact; the argument James produced would apply equally well to normal-sized and normally spaced wires containing many rows of idle wheels and cells. The argument runs like this.

Suppose the magnetic field is zero at first, and the switch open, so that all the cells and idle wheels are stationary (Figure 2a). When the battery is brought into the circuit by closing the switch, the idle wheels along AB move bodily from left to right without rotating. This causes the rows of cells on either side of AB to rotate in opposite directions, thus creating a circular magnetic field around the wire. The idle wheels in PQ are now pinched between rotating cells on the AB side and stationary ones on the other, so they start to rotate (clockwise) and also to move from right to left, the opposite direction from those in AB (Figure 2b).

But the circuit containing the wire along PQ has some resistance (all circuits do), so the idle wheels there, after their initial surge, will slow down, causing the cells above PQ to begin rotating anticlockwise. Soon, the sideways movement of the idle wheels will stop, although they will continue to rotate. By this time the cells above PQ will be rotating at the same rate as those in the row below PQ (Figure 2c).

When the switch is opened again, disconnecting the battery, the idle wheels along AB stop moving and the rows of cells on either side of AB stop rotating. The idle wheels in PQ are now pinched between stationary cells on the AB side and rotating ones on the other, so they start to move from left to right, the same direction as the original AB current (Figure 2d).

Once again the resistance of the circuit containing PQ causes the idle wheels there to slow down. This time, when their sideways movement stops they will not be rotating: we are back to the state represented in Figure 2a.

Thus, switching on a steady current in AB induces a pulse of current in PQ in the opposite direction and switching the current off induces another pulse in PQ, this time in the same direction as the original current. More generally, any change of current in the AB circuit induces a current in the separate PQ circuit through the changing magnetic field that links them. Equivalently, any change in the amount of magnetic flux passing through a loop of wire induces a current in the loop – effect 4 is explained. If the battery in the AB circuit were replaced by an a.c. generator, the alternating AB current would induce an alternating current in PQ. This is exactly the way transformers work in our electrical power supply systems.

And here, at last, was a mechanical analogy for Faraday’s electrotonic state. It was the effect at any point in the field of the angular momentum of the spinning cells. Like a flywheel, the cells would act as a store of energy, reacting with a counterforce to resist any change in their rotation; this took the form of an electromotive force which would drive a current if a conductor was present.

James had now explained three of the four effects. He had not yet found a way of using the model to account for effect 1, forces between electric charges, commonly called electrostatic forces. But he wrote up the results with full mathematical rigor in a paper called On Physical Lines of Force, which was published in the Philosophical Magazine in monthly installments: Part1 appeared in March 1861 and Part 2 was spread over April and May. Not wanting to be misunderstood, he was at pains to point out that his bizarre arrangement of whirling cells and idle wheels was merely a model:

He had failed to achieve a full theory and it was with a feeling of disappointment that he and Katherine left London to spend the summer vacation at Glenlair. But it was good to get back to the easy rhythm of country life, with its concerns over crops, plantations and farm animals, and to the fresh Galloway air. He had not planned to do any serious work on electricity and magnetism during the summer and took no reference books. But his thoughts ran on and an idea began to crystallize.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, or even as that which I would willingly assent to as an electrical hypothesis. It is, however, a mode of connexion which is mechanically conceivable, and easily investigated, and it serves to bring out the actual mechanical connexions between the known electromagnetic phenomena; so that I venture to say that anyone who understands the provisional and temporary nature of this hypothesis, will find himself rather helped than hindered by it in his search after the true interpretation of the phenomena.

It seemed a small idea at first. The material making up his little cells had to transmit the twisting forces internally so that each cell would rotate as a body. To do this without dissipating energy the material needed to have a degree of springiness, or elasticity. The idea grew. Could this elasticity be the source of the forces between electric charges which he had so far been unable to explain?

In conductors currents could flow because the electrical particles, the idle wheels, were free to move bodily in response to an electromotive force. Continuous currents could not flow in insulators because the particles were bound to their neighboring cells. But elastic cells would distort, allowing the particles to move a short distance. The distortion in the cells would cause a restoring force, like a spring. The particles would move until the spring-back force balanced the electromotive force.

So, for example, if a battery were connected by metal wires across two metal plates separated by an insulating material, there would be a small displacement of the electrical particles in the insulator away from one plate and towards the other. This small movement was, in effect, a brief electric current. The movement of electricity would be the same all round the circuit so that in the wires, where particles were not bound to cells, the same brief current would flow. This would result in a surplus of particles on the surface of one plate and a shortage on the surface of the other, so one plate would become positively charged and the other negatively. The distorted cells in the insulating material between the charged metal plates would act like a wound-up spring, exerting a mechanical force of attraction between them. So the elasticity of the cells explained the force between the charged plates.

Even when the battery was disconnected the spring would stay wound up, storing energy. If the plates were then connected by a metal wire, the spring’s energy would be released: a brief current would flow in the wire, the charges on the plates would fall back to zero and the cells and idle wheels would return to their rest positions.

Following his earlier thinking, James proposed that the elasticity, or spring-stiffness, of the cells would be modified if they were sharing their space with ordinary matter, and that the degree of modification would depend on the type of matter. The higher the electrical susceptibility of the substance, the softer the spring and the greater the electrical displacement for a given electromotive force. For example, filling the space between the metal plates with mica rather than air would soften the spring and increase the amount of charge on the plates for a given voltage across them.

He wrote up the mathematics and everything fitted together. James had shown how the electrical and magnetic forces which we experience could have their seat not in physical objects like magnets and wires but in energy stored in the space between and around the bodies. Electrostatic energy was potential energy, like that of a spring: magnetic energy was rotational, like that in a flywheel, and both could exist in empty space. And these two forms of energy were immutably linked: a change in one was always accompanied by a change in the other. The model demonstrated how they acted together to produce all known electromagnetic phenomena.

A triumph. But there was more. The model predicted two extraordinary and entirely new physical phenomena which took physics into undreamt-of territory.

One was that there could be electric currents anywhere, even in perfect insulators or in empty space. According to the model, as we have seen, there would be a little twitch of current whenever an electromotive force was first applied to an insulator, because the electrical idle wheels would move slightly before being halted by the spring-back force of their parent cells. In the model all space is filled with cells, so these twitches of current would occur even in empty space.

This new type of current would arise whenever the electric field changed. Its value, at any point, would depend on the rate of change of the electric field at that point. In fact, it was simply the rate of displacement of electricity due to the small movement of the particles. James asserted that it was in every way equivalent to an ordinary current. He gave it the name ‘displacement current’.

The equations governing electrical and magnetic effects had hitherto just dealt with the ordinary conduction current. In James’ new theory, the displacement current had to be added in. When this was done the system of equations was transformed from a motley collection into a beautifully coherent set. This was not immediately evident, however, even to James; he had seen something even more interesting.

All materials that have elasticity transmit waves. James’ all-pervading collection of cells was elastic, so it must be capable of carrying waves. In an insulating material, or in empty space, a twitch in one row of idle wheels would be transmitted via their parent cells to the surrounding rows of idle wheels, then to the rows surrounding them, and so on. Because the cells have inertia they would not transmit the motion instantly but only after a short delay – the twitch would spread out as a ripple. So any change in the electric field would send a wave through all space.

What is more, any twitch in a row of idle wheels would make the neighboring cells turn a bit and so generate a twitch in the magnetic field along the cells’ axes of spin. All changes in the electric field would therefore be accompanied by corresponding changes in the magnetic field, and vice versa. The waves would transmit changes in both fields; they were electromagnetic waves.

What kind of waves were they? Waves in the sea or along a rope are called ‘transverse’ because the individual particles of a sea or rope move at right angles to the direction of the wave. Waves like sound are called ‘longitudinal’ or ‘compression’ waves because the particles move back and forth along the same line as the wave. James’ electromagnetic waves are clearly transverse because the changing electric and magnetic fields were both at right angles to the direction of the wave.

James felt he was on the verge of a great discovery. Light waves were known to be transverse. Could light consist of waves of the kind his model predicted? The speed of light was known with reasonable accuracy from experiments and astronomical observations. It was also well known that the speed of waves in any elastic medium is given by the square root of the ratio of the medium’s elasticity to its density. In the model, the elasticity of the cells controlled the electrostatic (spring-back) forces and their density the magnetic (centrifugal) forces. James’ calculations showed that the spring stiffness of his cells in empty space was not completely determined: it could vary over a factor of three. But if he set it at the lowest value in this range – equivalent to assuming that the cell material was a hypothetically perfect solid – a remarkable result would follow. The wave speed in empty space, or in air, would then be exactly equal to the ratio of the electromagnetic and electrostatic units of electric charge . It seemed impossible that such a simple and natural result could be wrong, so James confidently set the elasticity of his cells to fit it.

He now had a very simple formula for the speed of his waves. To check their speed against that of light he first needed to look up the result of an experiment by Wilhelm Weber and his colleague Rudolf Kohlrausch. They had measured the ratio of the electrodynamic and electrostatic units of charge; electrodynamic units are closely related to electromagnetic ones, so James would easily be able to convert their result to give the value for the ratio he needed. He also needed to look up the exact values for the experimentally measured speed of light. But he had brought no reference books; this would have to wait until he got back to London in October. The summer passed in a glow of anticipation.

He had left for Glenlair disappointed at having failed to produce a complete theory of electromagnetism. He returned to London not only with a complete theory but with two entirely new predictions, displacement current and waves. Moreover, the waves might turn out to include light. He eagerly looked up Weber and Kohlrausch’s experimental result and from it worked the speed of his predicted waves. In empty space or air they would travel at 310,740 kilometers per second. Armand-Hippolyte-Louis Fizeau had measured the speed of light in air as 314,850 kilometers per second.

The correspondence seemed too close for coincidence, even allowing for a possible error of a few percent in each of the experimental results. Light must consist of electromagnetic waves. Some of the greatest leaps in science have come when two sets of apparently different phenomena are explained by a single new theory. This was one such leap: at a stroke, he had united the old science of optics with the much newer one of electromagnetism.

James had not expected to extend his paper On Physical Lines of Force beyond Parts 1 and 2 but now he set about writing Part 3, which covered electrostatics, displacement current and waves, and Part 4, which used his model to explain why polarized light waves change their plane of vibration when they pass through a strong magnetic field – an effect discovered experimentally by Faraday. Even for one with the calmest of temperaments it must have been an exhilarating time. The two further parts of the paper were published early in 1862. In Part 3, James announced:

The idea of a medium, an ‘aether’ pervading all space, was far from new. Physicists of the day believed an aether of some kind was necessary to transmit light waves, so one might have expected ready acceptance of James’ application of the principle to electricity and magnetism. But there were misgivings; the reaction of his friend Cecil Monro was typical:We can scarcely avoid the inference that light consists in the transverse undulations of the same medium which is the cause of electrical and magnetic phenomena.

The difficulty lay deep in the scientific thinking of the time. People believed that all physical phenomena resulted from mechanical action and that all would be clear to us if, and only if, we could discover the true mechanisms. With a century and a half of hindsight we can see the spinning cell model as a crucial bridge between old and new ideas – built from old materials but paving the way for a completely new type of theory, one which admits that we may never understand the detailed workings of nature. One cannot blame James’ contemporaries for seeing things differently. To many of them the model was simply an ingenious but flawed attempt to portray the true mechanism, for which the search would continue.The coincidence between the observed velocity of light and your calculated velocity of a transverse vibration in your medium seems a brilliant result. But I must say I think a few such results are needed before you can get peple to think that every time an electric current is produced a little file of particles is squeezed along between two rows of wheels.

James himself was not entirely content with the model, but for different reasons. He wanted to free the theory if possible from all speculative assumptions about the actual mechanism by which electromagnetism works. He was to achieve this wish 2 years later by taking an entirely new approach. Scientific historians now look upon his spinning cells paper as one of the most remarkable ever written but hold the one that followed to be greater still.