The Man Who Changed Everything

These are clearly the words of a man who has not found it easy to master grief and loneliness. Nevertheless, for someone of his buoyant spirit it was a good life. His Aberdeen job was important to him, not so much for the status it conferred as for the chance it gave to help young people gain useful knowledge. He loved Glenlair and there were still many improvements he wanted to make on the estate and in the neighbourhood. Most of all, he was still fascinated by the physical world and determined to find out all he could about it.

Saturn, with its extraordinary set of vast, flat rings, was the most mysterious object in the universe. How could such a strange structure be stable? Why did the rings not break up, crash down into Saturn, or drift off into space? This problem had been puzzling astronomers for 200 years but it was now getting special attention because St John’s College, Cambridge, had chosen it as the topic for their prestigious Adams’ Prize.

The Prize had been founded to commemorate John Couch Adams’ discovery of the planet Neptune. It may also have been an attempt by the British scientific establishment to atone for its abject performance at the time of the discovery. Adams had spent 4 years doing manual calculations to predict the position of a new planet from small wobbles in the movement of Uranus, then the outermost known planet, but his prediction, made in 1845, was ignored by the Astronomer Royal, Sir George Airy. The following year, the Frenchman Urbain Leverrier independently made a similar prediction. He sent it to the Berlin Observatory, who straightaway trained their telescopes to the spot and found the planet. Perhaps to soothe his conscience, Airy made a retrospective claim on Adams’ behalf. Some ill-mannered squabbling followed, from which the only person to emerge with credit was Adams, who had kept a dignified silence. In the end good sense prevailed; Adams and Leverrier were given equal credit. Adams later became Astronomer Royal.

The Adams’ Prize was a biennial competition; the Saturn problem had been set in 1855 and entries had to be in by December 1857. The problem was fearsomely difficult. It had defeated many mathematical astronomers; even the great Pierre Simon Laplace, author of the standard work La mécanique céleste, could not get far with it. Perhaps the examiners had set the problem more in hope than expectation. They asked under what conditions (if any) the rings would be stable if they were (1) solid, (2) fluid or (3) composed of many separate pieces of matter; and they expected a full mathematical account.

James tackled the solid ring hypothesis first. Laplace had shown that a uniform solid ring would be unstable and had conjectured, but could not prove, that a solid ring could be stable if its mass were distributed unevenly. James took it from there. Perhaps thinking ‘where on Earth can I start?’, he started at the centre of Saturn, forming the equations of motion in terms of the gravitational potential at that point due to the rings. (Potential in gravitation is roughly equivalent to pressure in water systems; difference in potential gives rise to forces.)

In an astonishing sequence of calculations, using mathematical methods which had been known for years but in unheard-of combinations, he showed that a solid ring could not be stable, except in one bizarre arrangement where about four-fifths of its mass was concentrated in one point on the circumference and the rest was evenly distributed. Since telescopes clearly showed that the structure was not lopsided to that extent, the solid ring hypothesis was despatched. James sent his friend Lewis Campbell a progress report, drawing on the Crimean war for his metaphors:

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 am now splash into the fluid one, amid a clash of symbols truly astounding. When I reappear it will be in the dusky ring, which is something like the siege of Sebastopol conducted from a forest of guns 100 miles one way, and 30,000 miles the other, and the shot never to stop, but to go spinning away round in a circle, radius 170,000 miles.

Could fluid rings be stable? This depended on how internal wave motions behaved. Would they stabilise themselves or grow bigger and bigger until the fluid broke up? James used the methods of Fourier to analyse the various types of waves that could occur, and showed that fluid rings would inevitably break up into separate blobs.

He had thus shown, by elimination, that although the rings appear to us as continuous they must consist of many separate bodies orbiting independently. But there was more work to do: the examiners wanted a mathematical analysis of the conditions of stability. A complete analysis of the motion of an indeterminately large number of different-sized objects was clearly impossible, but to get an idea of what could happen James took the special case of a single ring of equally spaced particles.

He showed that such a ring would vibrate in four different ways, and that as long as its average density was low enough compared with that of Saturn the system would be stable. When he considered two such rings, one inside the other, he found that some arrangements were stable but others were not: for certain ratios of the radii the vibrations would build up and destroy the rings. This was as far as he could go with calculation but he recognised that there would be collisions between the particles —a type of friction—and predicted that this would cause the inner rings to move inwards and the outer ones outwards, possibly on a very long time-scale.

James was awarded the Adams’ Prize. In fact, his was the only entry. This boosted rather than diminished his kudos because it demonstrated the difficulty of the task; no one else had got far enough to make it worth sending in an entry. The Astronomer Royal, Sir George Airy, was not, as we have seen, the most reliable judge of scientific merit but he was on safe ground when he declared James’ essay to be ‘One of the most remarkable applications of Mathematics to Physics that I have ever seen’. The work had been a Herculean labour. In fact it was a triumph of determination as much as creativity; the mathematics was so intricate that errors had crept in at every stage, and much of the time was taken up in painstakingly rooting them out. In all it was a marvellous demonstration of vision, intuition, skill and sheer doggedness and it earned James recognition by Britain’s top physicists; he was now treated as an equal by such men as George Gabriel Stokes and William Thomson.

Interestingly, no-one since Maxwell has been able to take our understanding of the rings much further. But flypast pictures from Voyager 1 and Voyager 2 in the early 1980s showed them to have exactly the type of structure that he predicted. Although the essay had won the prize, James spent a lot of time over the next 2 years developing it and trying to make it more intelligible to general readers before publishing it in 1859².

To demonstrate some of the kinds of wave motion that can exist in the rings, James designed a hand-cranked mechanical model and had it made by a local craftsman, John Ramage. When the handle was turned, little ivory balls mounted on a wooden ring could vibrate in two different modes. He said it was ‘for the edification of sensible image worshippers’. This was probably a dig at William Thomson, who used to say that the test of whether or not we understand a subject is ‘Can we make a mechanical model of it?’. The model is now kept in the Cavendish Laboratory in Cambridge, together with a beautiful ‘dynamical top’, also made by Ramage, which James designed to demonstrate the dynamics of a rotating body in illustration of a short paper on the subject. Clearly James was himself an admirer, if not a worshipper, of ‘sensible images’. The dynamical top was a great success as a teaching aid: James had several copies made for friends in various educational establishments and it became a commercial product which stayed in demand until the 1890s.

He took the top with him when visiting Cambridge to collect his M.A. and showed it to friends at an evening party in his room. They left it spinning and one of them was astounded to see it still going round when he called to get James out of bed in the morning. James had seen him coming, started the top and hopped back under the blankets.

Saturn had, for the time being, almost ousted James’ other research interests. But with the great labour of the Prize essay out of the way (it weighed 12 ounces), he turned with some relief to optics. He took his colour vision work further, using an improved design of colour box splendidly built by the invaluable Ramage and an ingenious new method of colour matching.

The old method was to find out by direct comparison what mixture of the three primary colours matched the fourth spectrum colour being investigated. Instead, he now mixed the colour under investigation with two of the three primaries and found out what combination of these gave a match with natural white light. By already knowing what combination of all three primaries made white, and using a little simple algebra, he could convert the results into the usual form. This method was simpler to operate and made for more consistent matching.

It also provided a neat new solution to the same problem he had faced when experimenting with the colour top—his primary colours inevitably failed to correspond exactly to the characteristics of the eye’s three receptors, and so he could not produce all colours by direct mixing. His earlier solution for the awkward colours had been to put a negative quantity of one primary into the mix by combining it with the colour to be matched, rather than with the other two primaries. But the new method gave a simple, direct match for all colours; sometimes this implicitly involved a negative amount of one primary but the negative sign appeared only when the relevant colour equation had been recast into the usual form.

He also put forward a new approach to the theory of optical instruments, defining the instrument by what goes into it and what comes out (what engineers call the ‘black box’ approach), rather than by the details of the internal reflections and refractions. He simply worked out the geometrical relations between the spaces occupied by the object and the image—in modern engineering terms, the transfer function between the instrument’s input and its output. Optics was a highly developed branch of science and yet no-one before Maxwell had thought of doing this.

James had become a favourite with the College Principal, the Rev. Daniel Dewar, and his family. He often visited their house and was asked to join them on a family holiday. He and Professor Dewar’s daughter, Katherine Mary, enjoyed being together and became more and more attracted to one another. This was, as far as we know, James’ first romantic attachment since the doomed affair with his cousin Lizzie. He proposed and Katherine accepted; they became engaged in February 1858 and were married in Aberdeen in June.

Lewis Campbell came up from Hampshire to be best man. He and his new wife were the Maxwells’ first guests at Glenlair. James had been best man at their wedding in Brighton a few weeks earlier. It was a joyful time. So much is clear from James’ letters to friends: the metaphors fly even more exuberantly than usual. Here he writes to Campbell in March to tell him of the engagement and the approximate date of the wedding, and to invite him and Mrs Campbell to Glenlair afterwards:

When we had done with the eclipse today, the next calculation was about the conjunction. The rough approximations bring it out early in June ...

The first part of May I shall be busy at home. The second part I may go to Cambridge, to London, to Brighton, as may be devised. After which we concentrate our two selves at Aberdeen by the principle of concerted tactics. This done, we steal a march, and throw our forces into the happy valley, which we shall occupy without fear, and we only await your signals to be ready to welcome reinforcements from Brighton ...

It was an unusual match. Katherine was 34, 7 years older than James, and may well have almost given up hope or thought of marriage before James came along. Both had known a measure of loneliness, and perhaps felt a joyful relief at having found a lifetime companion. James said as much in a poem:

Trust me spring is very near,
All the buds are swelling;
All the glory of the year
In those buds is dwelling.

What the open buds reveal
Tells us—life is flowing;
What the buds, still shut, conceal,
We shall end in knowing.

Long I lingered in the bud
Doubting of the season,
Winter’s cold had chilled my blood—
I was ripe for treason

Now no more I doubt or wait,
All my fears are vanished
Summer’s coming dear, though late,
Fogs and frosts are banished.

They both looked forward to being together at Glenlair. James had accompanied his proposal with a poem that invites Katherine to share the place that meant so much to him:

Will you come along with me,
In the fresh spring tide,
My comforter to be
Through the world so wide?

Will you come and learn the ways
A student spends his days
On the bonny bonny braes
Of our ain burnside.

And the life we then shall lead
In the fresh spring tide
Will make thee mine indeed,
Though the world be wide.
No stranger’s blame or praise
Shall turn us from the ways
That brought us happy days
On our ain burnside.³

Their honeymoon was a month of enjoying ‘sun, wind, and streams’ at Glenlair before James got back to work. Katherine helped where she could, particularly with the colour vision experiments, using the colour box. They each obtained matches of a range of pure spectrum colours with mixtures of red, green and blue and James included both sets of observations in his published results⁴. Plotted on a chromaticity diagram, they are very close to the standard results used today, which were published by the Commission Internationale d’Eclairage in 1931.

In April 1859, James read a paper by the German physicist Rudolf Clausius, which captured his imagination at once. It was about diffusion in gases—for example, the speed with which the smell from a bottle of perfume will spread through air when the bottle is opened. The eighteenth century Swiss physicist and mathematician Daniel Bernoulli had proposed what later became known as the kinetic theory of gases: that gases consist of great numbers of molecules moving in all directions, that their impact on a surface causes the gas pressure that we feel, and that what we experience as heat is simply the kinetic energy of their motion. Others had developed the theory further and by the mid-nineteenth century it was able to offer an explanation for most of the gas laws on pressure, volume and temperature which had been found by experiment.

But there was a difficulty over the speed of diffusion. For the theory to explain pressure at normal temperatures, the molecules would have to move very fast—several hundred metres per second. Why then do smells spread relatively slowly? Clausius proposed that each molecule undergoes an enormous number of collisions, so that it is forever changing direction—to carry a smell across a room it would actually have to travel several kilometres. The sheer scale of the process is astounding. James put it this way:

If you go at 17 miles per minute and take a totally new course 1,700,000,000 times in a second, where will you be in an hour?⁵

In his calculations, Clausius had assumed that at a given temperature all molecules of any one kind travel at the same speed. He knew this could not be exactly true but could not think what else to do. Nor, at first, could James. The problem of representing the motion of gas molecules mathematically was a bit like that he had faced with Saturn’s rings; and then he had settled for second best —calculating the results only for a simple special case. But now he had an inspiration which, at a stroke, opened the way to huge advances in our understanding of how the world works.

The standard mathematical methods, derived from Newton’s laws of motion, were of little use on their own because of the impossibility of analysing so many molecular motions, one by one. James saw that what was needed was a way of representing many motions in a single equation, a statistical law. He derived one, now known as the Maxwell distribution of molecular velocities. It said nothing about individual molecules but gave the proportion which had velocities within any given range.

The sheer audacity of the way he arrived at this law is astounding. It comes out in a few lines, with no reference to collisions; in fact there seems to be no physics in it at all. The argument goes something like this:

1. In a gas at uniform pressure and at a steady temperature, composed of molecules of one type, let the velocity of each molecule be represented by components x, y and z along three arbitrarily selected axes at right angles to each other. The total speed, s, of any particular molecule, irrespective of direction, will then be equal to the square root of the sums of the squares of its values of x, y and z (simply Pythagoras’ theorem extended to three dimensions and applied to the addition of velocities).

2. As the axes are perpendicular to each other, the number of molecules with any particular value of one velocity component, say x, will not depend on the numbers which have particular values of the other components y and z. But it will depend on how many molecules have particular values of the total speed s⁶.

3. Since there is no reason for molecules to move faster in any one direction than another, the form of the velocity distribution is the same along each axis.

These three statements imply a particular kind of mathematical relationship, which is easily solved, giving a formula for the statistical distribution of velocities along any of the three axes. Since the axes are arbitrary the velocities in any other direction also have this distribution.

This was the first-ever statistical law in physics—the Maxwell distribution of molecular velocities. The distribution turned out to have the bell-shaped form that was already familiar to statisticians and is now generally called the normal distribution. The top of the bell-curve corresponded to a velocity of zero and its sides were symmetrical in the positive and negative directions. Its shape varied with temperature: the hotter the gas, the flatter and wider the bell. The average velocity in any particular direction was always zero, whatever the temperature, but the average speed, irrespective of direction, was greater the higher the temperature. And from the statistical distribution of velocities, it was a simple matter to derive the distribution of speeds.

He had made a discovery of the first magnitude. It opened up an entirely new approach to physics, which led to statistical mechanics, to a proper understanding of thermodynamics and to the use of probability distributions in quantum mechanics. If he had done nothing else, this breakthrough would have been enough to put him among the world’s great scientists.

The key to the argument was the assumption, embodied in statement 2, that the three components of velocity are statistically independent. This was pure intuition. James felt that it must be true, although he conceded that the assumption ‘may appear precarious’. Years later the formula was verified in experiments, showing that his intuition was correct.

Like so many of James’ ideas, this one sprang from analogy. For years, physicists had used statistical methods to allow for errors in their experimental observations; they knew that errors in measurement tended to follow statistical laws. Social scientists, too, had used such methods to study characteristics of populations. What had occurred to no-one before Maxwell was that statistical laws could also apply to physical processes. James recalled reading, about 9 years earlier, an account of the work of Adolphe Quetelet, the Belgian statistician, which included a simple derivation of the formula for errors which underlies the method of least squares, a way of making the best estimate from a scattered set of observations⁷. This gave just the analogy he needed. In hindsight it seems so simple; anyone could have picked up the least squares formula and applied it to gases. But to make the connection it needed, to repeat Robert Millikan’s words, ‘one of the most penetrating intellects of all time’.

At this time, no-one knew that gases consist of molecules, still less whether it was their motion that determined their physical properties. Even among physicists who favoured the idea of molecules, most still held to Newton’s conjecture that static repulsion between them was the cause of pressure. Indeed, James had been taught the static theory at Edinburgh, but his intuition drew him strongly to the kinetic theory. It had by now become a plausible contender because it could explain physical laws which had already been found by experiment. But James went further: he used the kinetic theory to predict a new law. Now there could be a proper test: if experiments showed the prediction to be false, then the theory would be disproved, but if they showed it to be true, the theory would be greatly strengthened.

The new law that he predicted seemed to defy common sense. It was that the viscosity of a gas—the internal frictional that causes drag on a body moved through it—is independent of its pressure. One might expect a more compressed gas to exert a greater drag; even James was surprised at first that the theory said otherwise. But further thought showed that, at higher pressure, the effect on a moving body of being surrounded by more molecules is exactly counteracted by the screening effect they provide: each molecule travels, on average, a shorter distance before it collides with another one. A few years later, James and Katherine themselves did the experiment which showed the prediction to be correct.

This was James’ first venture into gas theory. It was a magnificent piece of work but by no means devoid of flaws. He made a second prediction—that viscosity should increase as the square root of absolute temperature—but, as we shall see, when he later tested this by experiment it turned out to be wrong. He made mistakes when trying to prove a relation which he was intuitively convinced was true: that the energy in a gas is equally divided between linear and rotational energy. His intuition was right—the principle is an important one now called the equipartition principle—but his proof was faulty. There were more mistakes in his derivation of equations for heat conduction. And he made some arithmetical errors: he was out by a factor of about 8000 when calculating the ratio of the thermal conductivity of copper to that of air because he had forgotten to convert kilograms to pounds and hours to seconds!

For all its faults, the work drew admiration, particularly from continental scientists. Clausius was prompted by it to make another attempt at some of the intractable problems, at the same time pointing out James’ errors. Gustav Kirchhoff, who is best remembered as the inventor of spectroscopy, said, ‘He is a genius, but one has to check his calculations’. Even these admirers failed to see the full significance of James’ introduction of statistical methods into physics. But there was one man who did. He was at this time a teenage student in Vienna and did not see James’ paper until about 5 years later, but was then so inspired by it that he spent much of a long and distinguished career developing the subject further; his name was Ludwig Boltzmann. During the 1860s and 1870s he and James took turns in breaking new ground, and Boltzmann continued after James died, putting the science of thermodynamics on a rigorous statistical basis. Although they never thought of themselves as such, they were a splendid partnership. It is fitting that their names are now immutably joined in the Maxwell-Boltzmann distribution of molecular energies.

James presented his results when the British Association for the Advancement of Science met in Aberdeen in September 1859, and published the paper in two parts the following year⁸. The meeting was a big event, attended by Prince Albert, and a number of interested citizens, including James, had raised the money for a fine new building to house it. Much later, the building became the town Music Hall and during the early 1900s a firm of advocates had the job of paying small dividends to the original subscribers. Not being able to trace one of them, they eventually put an advertisement in a local newspaper, asking anyone who knew the whereabouts of a Mr James Clerk Maxwell to get in touch. A school inspector who had made a study of Maxwell’s time in the town went along and asked if they had really never heard of Professor Clerk Maxwell, the most famous man ever to walk the streets of Aberdeen. No, they hadn’t. After the inspector had given a fulsome account of Maxwell’s accomplishments, the advocate said ‘That’s very interesting. We put the advertisement in because for years we have been sending dividends to Mr James Clerk Maxwell, Marischal College, and they have always been returned “not known”’.

What of James’ classes? Had he succeeded as a teacher? The nearest short answer one can give is ‘no’ and ‘yes’. For all his talents, he never mastered the technical part of teaching. He would prepare a lesson beautifully, do fine for a time while he stuck to his script, and then fly into analogies and metaphors which were intended to help the students but more often than not mystified them. He was not expert on the blackboard, where he made algebraic slips which took time to find and correct. And yet the students liked him and some found him truly inspiring. This report is from George Reith, who became Moderator of the Church of Scotland and father of Lord Reith, the first Governor of the British Broadcasting Corporation:

But much more notable [than the other professors] there was Clerk Maxwell, a rare scholar and scientist as the world came to know afterwards; a noble-souled Christian gentleman with a refined delicacy of character that bound his class to him in a devotion which his remarkably meagre qualities as a teacher could not undo.

And this one from David Gill, who became Director of the Royal Observatory, Cape of Good Hope:

After the lectures, Clerk Maxwell used to remain in the lecture room for hours, with three or four of us who desired to ask questions or discuss any points suggested by himself or ourselves, and would show us models of apparatus he had contrived and was experimenting with at the time, such as his precessional top, colour box, etc. These were hours of the purest delight to me.

... to those who could catch a few of the sparks that flashed as he thought aloud at the blackboard in lecture, or when he twinkled with wit or suggestion in after lecture conversation, Maxwell was supreme as an inspiration.

It seems paradoxical that such a fine scientific writer should be so lacking in basic teaching skill, especially as he believed fervently in the value of good education and had firm ideas on the principles to be followed. The principal difficulty lay in oral expression. It did not arise when he gave a formal speech and stuck closely to his text, nor in ordinary conversation, where, in congenial company, he could give free rein to his imagination. But in the lecture room he seemed to be caught between the two modes. Appreciating that people learn in different ways, he may have tried too hard to bring in helpful illustrations and analogies, confusing his audience with a welter of rapidly changing images. In Lewis Campbell’s words, ‘the spirit of indirectness and paradox, though he was aware of the dangers, would often take possession of him against his will’.

And perhaps he was too much of an idealist. All good teachers aim, as he did, to teach people to think for themselves, but most also recognise that all some students want is to gain a secondhand smattering of the subject so that they can pass exams, and make a specific effort to help them succeed in this limited ambition. Maxwell never did.

He did, however, do his utmost to help any student who truly wanted to learn. Students could take only two books at a time from the library but professors could take any number, and sometimes took a book for a friend. James used to take out books for his students, and when challenged by colleagues replied that the students were his friends. And his talks to working men were remembered long after he left Aberdeen. A farmer recalled how the professor had stood one of them on an insulating mat and ‘pumpit him fu’ o’ electricity’ so that his hair stood on end.

The debate about merging the two universities had moved on. The Royal Commission had decided that they would merge and the issue became whether there should be ‘union’, a common management of otherwise little-changed faculties, or ‘fusion’, a complete amalgamation, which would halve the number of professors. The fusionists gained the day and the new University of Aberdeen was set to get under way at the beginning of the academic year 1860-1861. There would be only one chair in natural philosophy, and James’ rival for the post was David Thomson, his opposite number at King’s College. Thomson was also Sub-Principal and Secretary of the College and an astute negotiator who had earned the nickname ‘Crafty’. James was clearly up against it, especially as to discharge him was the cheaper option—he would get no pension, not having served the requisite 10 years. Set against these factors was James’ achievement in research, but only a few people had any idea of its importance and none of them lived in Aberdeen. Thomson was chosen and James was made redundant.

Exactly at this time, James Forbes’ post as professor of natural philosophy at Edinburgh University became vacant—Forbes had been ill and was leaving to become Principal of St Andrews University. This was an appealing possibility. James would be succeeding his great friend and mentor. And in Edinburgh he would be among friends and relations and relatively close to Glenlair. He applied and so did his friend P. G. Tait, who was still at Belfast. This time the Aberdeen order was reversed and Tait was preferred.

There was plenty to do in the meantime. Besides preparing his great paper on gas theory for publication, he wrote another, on elastic spheres, and sent a report on his colour experiments to the Royal Society of London. Shortly afterwards he heard that the Society had recognised this work by awarding him the Rumford Medal, its highest award for physics. There was estate business to see to and an important local project was to raise funds for the endowment of a new church at Corsock, just north of Glenlair.

During the summer he went to a horse fair and bought a handsome bay pony for Katherine. Soon after returning he became violently ill, with a high fever. It was smallpox, almost certainly contracted at the fair, and it nearly killed him. James was in no doubt that it was Katherine’s devoted nursing that saved his life.