Professor Maxwell's Duplicitous Demon

by Brian Clegg

  It is desirable that an attempt should also be made to determine on which of the above hypotheses the appearance of both of the bright rings and the recently discovered dark ring may be most satisfactorily explained; and to indicate any causes to which a change of form, such as is supposed from a comparison of modern with the earlier observations to have taken place, may be attributed.

  Taken in isolation, the rings of Saturn provided an odd topic for Maxwell to work on. Although he did have wide interests in physics, he had never shown any great enthusiasm for astronomy beyond a youthful appreciation of the glories of the night sky from Glenlair, where a lack of street lighting had made for excellent star viewing. The same would be true for the rest of his career. It has been suggested that rather than reflecting the same kind of interest as he clearly had in colour, gases and electromagnetism, entering the competition was merely to take on the challenge of a puzzle to be solved, with the added bonus of a noteworthy prize attached.

  Maxwell duly applied his mathematical expertise to the problem of the rings, an oddity that had challenged many of the greats. The first to recognise the rings as such, the Dutch scientist Christiaan Huygens, thought that they consisted of a single solid flat structure, but as telescopes became better able to resolve their detail it was increasingly clear that there were multiple rings surrounding the planet. The dark band that appeared to be a gap dividing two rings, referred to in the prize challenge, was first noticed in 1675 by Italian astronomer Giovanni Cassini, though strictly speaking the gap could have been a dark part of the same structure.

  In 1787, French mathematician Pierre-Simon Laplace had got as far as demonstrating mathematically that the rings could not be continuous, symmetrical solid structures, as no material would be strong enough to prevent gravitational forces from ripping them apart. As Maxwell pointed out in his prize entry, iron, for example, would not only be plastic under the gravitational stress but would partially liquefy. Even if the rings rotated, which would partly counter the gravitational forces trying to tear them apart, the motion would fail to stabilise the situation, as such a model required inner parts of the ring to move faster than outer parts – the opposite of the reality of a single solid ring.

  Instead, Laplace suggested that it was possible to make solid rings stable if the mass in them was unequally distributed. However, one of Maxwell’s achievements in his Adams Prize entry was to show that, while Laplace was correct, the only stable structure would be like an engagement ring featuring an immense diamond, with 80 per cent of its mass concentrated on one spot. Such an uneven distribution of matter would have been clearly visible through the telescopes of the day. It just wasn’t the case.

  Next, Maxwell considered the possibility that what appeared to be solid rings were in fact bands of liquid, a kind of space river that surrounded the planet. As he started work on the more complex mathematics involved in dealing with fluid dynamics, he wrote to his friend Lewis Campbell:

  I have been battering away at Saturn, returning to the charge every now and then. I have effected several breaches in the solid ring and now I am splash [sic] into the fluid one, amid a clash of symbols¶ truly astounding.

  Here, Maxwell brought into play a mathematical tool that had been in use in an ad-hoc way for centuries, but had only been made generally applicable in the nineteenth century: Fourier analysis. The name refers to French mathematician Joseph Fourier, who in 1807, in a paper on the transfer of heat through solid bodies, showed that it was possible to break down any continuous function – effectively anything that could be represented on a continuous graph, however strangely shaped – into components that were simple, regularly repeating forms such as a sine wave.

  It might seem unlikely but, for example, even a ‘jerky’ function such as a square wave can be broken down this way, provided we are allowed to use an infinite set of components to make it up exactly (see Figure 2).

  Physicists and engineers now make use of Fourier analysis as a matter of course, but in Maxwell’s day the technique was still something of a novelty. Yet by using it, he showed that the way waves would combine should there be any disturbance in the rings (which would inevitably arise due to the gravitational attraction of Saturn’s moons and of Jupiter) made it impossible for a fluid to be as continuous as the rings appeared to be – they would simply not be stable, and the liquid or gas would accumulate in large globules. In effect if the rings were fluid, Saturn would end up with blobby moons.

  With solid rings and fluids dismissed, Maxwell deduced that the rings were most likely collections of vast numbers of small particles, held in place by gravity – but that the distance between us and Saturn meant that we could not make out the individual particles, an analysis that has stood the test of time and close-up examination of the rings. His approach involved exploring the mathematics of displacements in a series of small satellites in the same orbit to see how waves caused by any disturbances could travel without breaking the ring apart.

  FIG. 2. As more simple waves are added in, the result gets closer and closer to a square wave.

  Maxwell’s conclusion: ‘The final result, therefore, of the mechanical theory is, that the only system of rings which can exist is one composed of an indefinite number|| of unconnected particles revolving around the planet with different velocities according to their respective distances.’ Not only was this the only stable conclusion, in keeping with the second part of the question it was also the only one that could sensibly explain the multiple ring structure.

  Maxwell began his essay with the words ‘E pur si muove’ – the phrase that was allegedly, though almost certainly not, said by Galileo under his breath after recanting at his trial for heresy. Meaning ‘And yet it does move’, it seems a slightly odd choice in relation to the topic. Candidates were obliged to begin their essay with a motto, as a separate document linking the candidate’s name to the motto was then used to preserve the anonymity of the essays until after the judging took place. It’s possible Maxwell chose the words in honour of Galileo’s discovery of the rings (or, at least, that Galileo noticed there was something odd about Saturn).

  As it happens, the motto proved unnecessary. Maxwell’s was the only entry that year and, not surprisingly, he won the prize. Astronomer Royal George Airy remarked that this was ‘One of the most remarkable applications of Mathematics to Physics that I have ever seen’. Rather than just stop with his submitted essay, Maxwell took into account some comments made by the judging panel and developed his work as a background activity over the next few years, including having a mechanical model built by the Aberdeen-based instrument maker John Ramage, showing how waves could travel if the rings were made up of 36 satellites (made of ivory in the model). Obviously, reality would involve a far greater number, but the model demonstrated how fast-moving waves would be transmitted through the system. This was all before publishing the final version of his work.

  Maxwell’s prize entry may sound like an impressive but limited piece of work – the dynamics of the rings of Saturn appear to be something of a one-off application – but like many of his thought excursions throughout his working life, it would continue to have ramifications that went far beyond the initial use. The formation of planetary systems, such as our solar system, is a complex and not fully agreed problem to this day, but the best accepted theory depends on the accumulation of a disc of gas and dust, and this owes much to Maxwell’s work on Saturn’s rings. His achievement has been commemorated in a small way by naming a break in Saturn’s C ring the ‘Maxwell gap’.

  This would not be the last time that Maxwell’s unusual (for the day) combination of mathematics and physics would astound his contemporaries. It’s arguable that Maxwell was the most significant player in the transforming of physics into its modern form, from a largely descriptive discipline to one where mathematics began to drive progress in the field. It’s not that earlier physicists had ignored maths. Newton famously, for example, devised calculus so that he could perform his work on
gravity. However, Maxwell took things further, moving from simply explaining observations through mathematics to building mathematical models that took on a life of their own.

  Life in Aberdeen

  Meanwhile, Maxwell was not content with his teaching work at the university and, much as he had been involved with the Working Men’s College at Cambridge, he gave an evening class as a contribution to the Aberdeen School of Science and Art. This was a body devoted to educating tradesmen and others with day jobs that prevented them from attending daytime lectures. As we have seen, this kind of paid evening education, often linked to bodies such as mechanics’ institutes, was common in the nineteenth century and helped many improve themselves. Students, who would typically already have worked for twelve hours from 6.00am, paid 8 shillings** a year for 24 lectures spread over five months from November to April. The School of Science and Art had no premises for Maxwell’s classes (though it was able to use the library of the Mechanics’ Institute), but he was allowed to give his lectures using the facilities at Marischal College.

  It might seem that Maxwell had an overwhelming amount of work on, but he still found time for socialising, and for an activity that, as we have seen, he kept up throughout his life – writing verse. In 1857, Maxwell’s friend William Thomson was largely engaged with the laying of the transatlantic cable to send telegraph signals between the UK and the US. In a letter to Lewis Campbell, Maxwell noted that he had sent ‘great screeds’ to Thomson about Saturn’s rings, but it turned out that he was busy ‘a-laying of the telegraph which was to go to America’. In the process the cable broke. (According to Maxwell, this was because Thomson was ‘bringing his obtrusive science to bear upon the engineers, so that they broke the cable with not following (it appears) his advice’.)

  Maxwell includes in his letter the words of a song he ‘conceived on the railway to Glasgow’. He notes that to avoid ‘vain repetitions’, ‘let (U) = “Under the sea”’, so that ‘2(U), by parity of reasoning, represents two repetitions of that sentiment’. The first two verses of ‘The Song of the Atlantic Telegraph Company’ (with an entertaining prefiguring of a song from Disney’s The Little Mermaid) read as follows:

  2(U)

  Mark how the telegraph motions to me,

  2(U)

  Signals are coming along,

  With a wag, wag, wag;

  The telegraph needle is vibrating free,

  And every vibration is telling me

  How they drag, drag, drag,

  The telegraph cable along.

  2(U)

  No little signals are coming to me,

  2(U)

  Something has surely gone wrong,

  And it’s broke, broke, broke;

  What is the cause of it does not transpire,

  But something has broken the telegraph wire

  With a stroke, stroke, stroke,

  Or else they’ve been pulling too strong.

  Although Saturn had been an entertaining distraction, Maxwell had not abandoned the work on colour vision that he had started at Cambridge. His colour top to see the effect of combining different colours was useful, but it was relatively poorly calibrated. He was still writing papers based on it, as witness an enthusiastic letter from Michael Faraday, who wrote to Maxwell about receiving papers on the top and on Faraday’s lines of force:

  I have just read and thank you heartily for your papers. I intended to send you copies of two of mine. I think I have sent them, but do not find them ticked off. So I now send copies, not because they are assumed as deserving of your attention, but as a mark of my respect and desire to thank you in the best way that I can.

  Receiving such a letter from one of his scientific heroes must have thrilled Maxwell.

  At Aberdeen, however, he was to move beyond the colour top. With the help again of instrument maker John Ramage, he constructed a ‘light box’ in which adjustable-width slits could be used to vary the amount of each of the primary colours of light, beams of which were sent into a rectangular box where they were focused with a lens to produce a combined colour.

  This was not just a matter of observing the behaviour of light – it also enabled Maxwell to continue his exploration of the nature of colour vision, and particularly the origins of colour blindness. In 1860, he would be awarded the highly prized Rumford Medal of the Royal Society for his work on colour vision, and he would continue working with light boxes in London after he left Aberdeen.

  His achievements on Saturn and colours were undoubtedly small triumphs for Maxwell – but neither proved to be his most significant work while at Aberdeen, being little more than a pair of amusing diversions. Far more would rest on his development, starting in 1859, of the kinetic theory of gases. Although from the twentieth century onwards, Maxwell’s chief claim to fame would be his work on electromagnetism,†† at the time of his death, his work on gases was seen as the highlight of his career, as the electromagnetic theory was not yet widely understood.

  E pur si muove

  The term ‘kinetic theory of gases’ is misleading. The theory has much to say about the nature of heat and of matter. It would be far more meaningful if it had been called the statistical theory of gases, not only because it has statistics at its heart, but also because it was the forerunner of the many ways that statistics would become fundamental to science. However, the ‘kinetic’ term was useful at the time to distinguish the new theory from the prevailing ‘static’ theory, which, rather than realising as we now do that molecules are zooming around and colliding, assumed that they sat around, busily repelling each other, producing the force of gas pressure from this mutual repulsion.

  Statistics was originally a term for data about a country – it described a state, hence the name – but by the eighteenth century it was transformed by the gradual incorporation of the then new-fangled probability theory to become a mechanism for predicting behaviour of systems based on the likely behaviour of their components.‡‡ And gases proved the ideal application for this approach. A gas had been regarded as a fuzzy elastic medium where the constituents somehow repelled each other so that the gas would fill the available space. But the German physicist Rudolf Clausius had started a new way of thinking about gases, regarding their behaviour as the result of the interplay of vast numbers of randomly moving, colliding bodies, which could only be treated statistically.

  Statistics struggles with individuality§§ or with systems that are chaotic, in the mathematical sense of having an interplay between highly interacting influences, resulting in large unpredictable variations over time as a result of small initial differences. But a gas is made up of large numbers of identically behaving components with no individual ‘personalities’ and only a small number of factors such as temperature and density influencing their behaviour, making it ideal for statistical analysis.

  As far back as the eighteenth century, the mathematician Daniel Bernoulli had made the first use of statistical theory on gases, and in Maxwell’s time, for those who believed in atoms, many of the behaviours of gases were understood this way. We don’t know what a single molecule of gas is going to do. But we do know, given the many billions of molecules present in a room, how their behaviour will average out. And it had gradually been realised that their movement had something to do with heat.

  Before Maxwell’s time there had been competing theories about the nature of heat. As we have seen, one approach described it as a kind of invisible fluid, ‘caloric’, that could move from one body to another. Much of the early work on heat engines and the conservation of energy was done using the caloric theory. But reflecting on the relationship of temperature and pressure in gases – and suspecting that pressure was due to the impact of gas molecules on the container – physicists began to suspect that caloric was an unnecessary complication, because what was being measured in reality was the energy with which the molecules that made up matter were jiggling about.

  It was suggested by the developers of this new ‘thermodynamic’ theory
, notably Clausius and Maxwell’s friend William Thomson, that temperature was a statistical measure of the energy of the molecules – the faster they move overall, the higher the temperature of the gas – while pressure reflected the impact from gas molecules hitting the walls of the container. But one of the behaviours of gas had yet to be explained using a statistical viewpoint (it would be left to Maxwell to take such a numerical approach) – this was the speed that a smell promulgated.

  When a smelly substance – whether a perfume or something less pleasant – is released into the air, it takes some time for the odour to reach a distant nose. Yet at room temperature, gas molecules in the air should be flying around at hundreds of metres per second. So why do smells not arrive almost immediately? Clausius pointed out that the problem with the expectation of high-speed odour delivery was that the scent molecules did not have a free and easy journey from the source to the nose. There were vast numbers of air molecules in the way. The odour’s progress across a room would be a bit like attempting to take a straight-line stroll across the centre of a dodgems¶¶ ride – you would soon be knocked over. Similarly, collisions of the scent molecules with other molecules would inevitably ensue, knocking the molecules in all sorts of different directions and slowing progress from A to B.

  Imagine a single odour molecule from a newly brewed pot of coffee starting on its way across the kitchen. If there were no obstacles in its path, it could cross the few metres in a fraction of a second. But in reality, it may well undergo so many collisions and changes of direction that it would have to travel several kilometres before it reached the nose of a waiting breakfaster. Although Clausius deduced from this model that there should be an average distance a molecule travels between collisions (known as the mean free path), he was not able to calculate this distance, a value that would have enabled him to get a feel for the actual size of molecules.