## Abstract

Since 1890, James Clerk Maxwell's reputation has rested upon his theory of electromagnetism. However, during his lifetime he was recognized ‘as the leading molecular scientist’ of his generation. We will explore the foundation of his significance before 1890 using his work on the stability of Saturn's rings and the development of his kinetic theory of gases, and then briefly discuss the grounds for the change of his reputation.

## 1. Introduction

In 1931, at the centenary of Maxwell's birth, Albert Einstein noted the significance of the former's theory of electromagnetism to physics and physicists at the beginning of the twentieth century (Harman 1998). Into the twenty-first century, his work on colour theory and electromagnetism remains equally important in key theoretical domains in physics and in the work of engineers. In contrast, when James Clerk Maxwell died in 1879, he was remembered, in the words of his old friend and colleague Peter Guthrie Tait (1831–1901), as ‘the leading molecular scientist’ of his generation (Garber *et al*. 1986). To understand the range of Maxwell's work and influence, we need to consider Tait's phrase ‘leading molecular scientist’ seriously and explore Maxwell's reputation and importance in physics during his lifetime. Tait and Maxwell met and became friends at the Edinburgh Academy in the 1840s and remained close throughout the latter's life. Their correspondence was frequent and informal, some of which in the form of postcards. Maxwell would hail Tait as O *T*′, as he would hail William Thomson as O T, and sign himself JCM or d*p*/d*t*. These came from Tait's expression JCM=d*p*/d*t* for the second law of thermodynamics. d*p*/d*t* identifies Maxwell on the plinth of his bust in the portrait gallery of Marischal College. In the 1870s, the expression of the second law and its meaning were contentious subjects and Tait published a history of thermodynamics that championed British physicists’ claims to discovering the first and second laws. While not content with Kelvin's formulation of the second law, Maxwell understood its significance only after having read the early papers of Josiah Willard Gibbs. He then introduced the work of Gibbs to Britain. He also published a textbook on thermodynamics that went through several editions in the 1870s (Maxwell 1872–1875, 1875*a*,*b*; Garber 1969).

## 2. Aberdeen and Cambridge

To trace the origin of Maxwell's work on molecules, we have to consider his early career and his arrival at Aberdeen in 1856. In his baggage on that trip north, Maxwell carried three pieces of research. The most complete was his first theoretical and experimental work on colour (Everitt 1975; Harman 1998); the second, the nearly completed paper ‘On Faraday's lines of force’ (Maxwell 1855–1856); and the third, which he had just begun, ‘Essay on Saturn's rings’. They were in disparate realms of physical theory, yet they all displayed the mathematical principles and practices he had developed at Cambridge in the previous three gruelling years. The experimental foundations of his work were from his apprenticeship in his own laboratory set up in a cottage on his family estate of Glenlair in Galloway and encouraged by James David Forbes (1809–1868) while he was a student at Edinburgh University.

At Aberdeen Maxwell focused his research energies on Saturn's rings, the subject of the 1855 Adams Prize essay. The Adams Prize competition, on a subject in pure mathematics, astronomy or natural philosophy, was announced once every 2 years and was open to all who had been admitted to a degree at Cambridge (Maxwell 1859). The two major examiners for this competition were William Thomson (1821–1907) and James Challis (1803–1882). Without considering the planet's satellites, the candidates had to examine the stability of Saturn's rings, assuming them to be concentric with the planet. They could be solid, liquid or ‘in part aeriform’. In the 1850s, Saturn and other planets were scrutinized by telescopes as they had been for some 200 years. The complex, yet predictable, path of a planet was still its most characteristic feature. Hence the emphasis on stability in the setting of the Adams Prize question. In addition, the contestants had to take into account the third, inner ‘dusky’ ring, recently discovered by the American astronomer George Bond (1825–1865) at Harvard. They had 2 years to complete their investigations (Brush *et al*. 1983).

Drawn from the latest available astronomical observations, the image of Saturn was of a bright white, marble smooth body, around whose equator were two flat rings. If you had a telescope as fine as that of George Bond, you could discern the third ring inside the other two, and the separations within the inner ones and the division of the outmost ring into two sets of thin rings. The Adams Prize question was a timely subject on a current theoretical puzzle in astronomy.

It was also just the subject for a competition to honour the work of the still active astronomer John Couch Adams (1819–1892). Honouring a living scientist was still quite extraordinary; but then so was Adams. In 1845, using his formidable mathematical skills he had postulated the existence of the planet Neptune, although he was happy to acknowledge Urbain Jean Joseph Leverrier's priority in actually sighting it. There are other instances of his lack of ambition and absolute control of the intricate mathematics necessary for theoretical planetary astronomy. The 1855 prize questions echoed the kinds of problems he tackled in his career and recognized ‘his humour, modesty and grace’, as his colleagues at St John's College put it in 1848 when establishing the prize in his name.

To understand Maxwell's approach to the Saturn's rings problem, we need to consider his education at Cambridge in the early 1850s. At the age of 20 Maxwell entered Cambridge rather later than most undergraduates. He first attended the University of Edinburgh where the professor of natural philosophy, James Forbes, recommended him to join Cambridge. Both Forbes and his mathematics coach at Cambridge, William Hopkins (1792–1866), agreed that Maxwell could not think incorrectly, physically, but his abilities in mathematics were far less than those needed to gain honours. Forbes wrote to William Whewell (1794–1866) of Maxwell's ‘exceeding uncouthness’ in mathematics and that the ‘drill’ of Cambridge was the only chance of taming him (Harman 1998).

In the 1850s, gaining high honours at Cambridge was so competitive that ambitious students needed the training of a private mathematics coach such as Hopkins 5 days a week over 2 years and during the summer before the contest in the Senate House the following January. A recent change of the regulations by the University Senate had lengthened the examination to a total of 8 days. By the time an undergraduate would sit for the tripos, his probable place was already surmised by his college tutor and his other teachers. He was either a ‘Pass’ man or one who could attain honours. Within the latter group were those who were expected to gain distinction in the final examination. The ‘Pass’ men sat only for the first 3 days of the Senate House examination. Those who passed the test in the elementary parts of mathematics with honours were admitted to a further 5 day examination on the higher parts of mathematics. The examination consisted of a series of mathematical questions of increasing difficulty in algebra, geometry, the calculus and Fourier analysis, often embedded in questions on mechanics, hydrodynamics and the theory of light. Quantity counted. The trick was to correctly answer as many of these questions as possible within the limited time of the examination. ‘Tough regimes of competitive technical training’ is the description of one historian (Warwick 2003).

Historians do not usually associate high-level mathematics with mid-nineteenth-century Cambridge. This is because not much of it was published. Cambridge was extraordinarily insular. Work that faculty might, if elsewhere, have published as research appeared as a tripos or a Smith Prize question. Stokes’ theorem appeared as a problem on the Smith Prize paper in 1854, set and administered by George Gabriel Stokes (1819–1903) and answered by Maxwell. We still have Maxwell's solution. The Smith Prize was a further gruelling 1 day examination on higher mathematics for those in the first five or so places in the tripos examination. Robert Smith (1689–1768) had established this prize as a corrective to favouritism in the tripos results. Examiners for the tripos were drawn from the fellows and faculty of the colleges. The larger colleges had a proportionally larger representation among them. Smith was suspicious of the overwhelming number of honours students from the same colleges. The Smith Prize was an antidote to such favouritism (Barrow-Green 1999).

Despite the rigour and intensity of the system, the students had recourse to old examination questions and hand-copied notes. In addition, private coaches had to keep their teaching abreast of examiners’ expectations. Lastly, there was the *Cambridge and Dublin Mathematical Journal*, established in 1837, to print problems and discuss techniques likely to be useful for the examinations, as well as to publish research. Natural philosophy courses were also mandated for undergraduates. The lectures were lightened with experimental demonstrations, but not bolstered by any trips to a laboratory. Teaching laboratories were only just appearing in Britain (the first was Kelvin's laboratory at the University of Glasgow established in the 1850s). Maxwell's skills in experiment were from his own efforts in his private laboratory and were unique in 1850s Cambridge.

## 3. Maxwell on Saturn's rings

As a young, recent Cambridge graduate (1854), who was placed second in the tripos, then first in the Smith Prize competition, Maxwell still had to consolidate his reputation. His work on Saturn's rings completed at Aberdeen did just that, something neither his work on Faraday's lines of force nor that on colour accomplished in his lifetime. He began his research on Saturn's rings with the obvious theoretical work on the subject, Pierre Simon, Marquis de Laplace's, four-volume magnum opus, *Traité de Mécanique Céleste* (1799–1825). The pattern of Laplace's work on physical or astronomical problems was to develop the mathematics abstractly, sometimes with many arbitrary functions and constants, and then simplify the expressions in the face of known data. His four-volume treatise was the most complete, and important, mathematical investigation of the heavens. It was the reference Maxwell had to consult and contradict only with very good reasons.

In his investigation of Saturn's rings, Laplace considered the stability of a single solid ring. He found that it would collapse into the planet if it was at rest. To meet the observed stability of the system meant that solid rings revolved about Saturn. Laplace also gave upper limits on their density. By 1855, in the light of recent observations, the structure and stability of Saturn's rings needed revisiting. Any investigation would require the exercise of intricate mathematics tied to sure-footed physical arguments.

In his Adams Prize essay, Maxwell first disposed of Laplace's argument for the stability of the rotating solid ring. It was incomplete. He found several types of solid rings unstable. For stability, this ring had to be weighted at one point by a mass approximately 41/2 times that of the ring. As Maxwell remarked, such a mass would surely be obvious to astronomers and none had been reported. In addition, a slight change in the load or in its position would shatter the ring. The problem of a liquid ring was that perturbations would tend to break the liquid into drops; these would accumulate and destroy the ring's stability.

He was left with rings made up of single ‘satellites’ of equal size revolving about Saturn and found that considerations of stability severely limited the masses of the satellites. Maxwell reduced the rings to one ring of satellites. Any satellite could be displaced from its mean positions radially, normally and tangentially to the ring. Instead of tackling the impossible task of dealing with their displacements individually, Maxwell also used Fourier analysis to show the propagation of waves in such a ring, which, under certain conditions, would not lead to collisions between the satellites and thus were stable. He used Fourier analysis to express the displacements of a single satellite as functions of time. He found that the tangential displacement was enhanced by the attraction of the satellite towards which it was moving. If at the same time the satellite was moving radially outwards, it would fall behind other members in the ring. The result was that the satellite would fall behind the others, its motions would be retarded and it would move inward again. The radial and tangential motions were coupled. Small disturbances normal or tangential to the plane of the ring would not disrupt it but lead to waves being propagated around it. Under the condition of stability these coupled motions led to four different kinds of waves each with its own velocity traversing the circumference of the ring. He then considered the motions of the satellites under unstable conditions.

However, Saturn has rings and he turned to the case of two such rings rotating about Saturn at their appropriate speeds. Maxwell found that in most cases their motions were stable, except for certain ratios of their radii. Then, the resonant waves would grow indefinitely until they would break up and their satellites would fly off in all directions colliding with each other in the process. Under stable conditions the two satellite rings produce eight different kinds of waves of different frequencies propagated around such rings.

Maxwell noted difficulties in taking the analysis further. For certain ratios of the radii of the rings, a wave of one type in one of the rings would come into resonance with a wave of the other type in the other rings. These would grow indefinitely and the rings ‘will be thrown into confusion’ and the satellites ‘would fly off in all directions and collide with members of other rings’. He understood that he could not describe the motions of a ring, or rings, made up of randomly moving particles. He also understood that all that he could do was examine the stability of a ring or systems of rings of particles under various disturbing forces.

He concluded by noting the inability of dynamics to address this last problem: ‘When we come to deal with collisions among bodies of unknown number, size, and shape, we can no longer trace the mathematical laws of their motion with distinctness’. Mechanics cannot deal with collisions among many bodies flying around randomly. All he could do was gather the possible scenarios for stability and instability.

Maxwell did not consider a ring made up of independent particles, something that he is sometimes assumed to have done. He had shown the circumstances under which waves would produce collisions between the particles. However, we must remember that Maxwell's object was none other than to demonstrate the conditions for stability. Within the limits of dynamics, Maxwell concluded that the only possible structure for Saturn's rings was concentric rings of satellites, each revolving with an appropriate speed. They would act on one another and produce perturbations in their motions. He also demonstrated the existence of conditions under which the motions of one or both rings became unstable. To further illustrate this system Maxwell had a model, constructed by the Aberdeen instrument makers Smith and Ramage, to demonstrate the motions of a ring of 36 satellites, which he used to illustrate his results on satellites’ motions (Brush *et al*. 1983).

## 4. Maxwell and kinetic theory

Maxwell was not done with the problem. To understand how he returned to these issues, we need to examine the early history of gas theory. In 1857, Rudolph Julius Emmanuel Clausius (1822–1888) published the first in a series of papers on gases (Clausius 1857). He was already known for his work in thermodynamics. However, in his papers on heat theory Clausius avoided any speculations about the structure of matter. His first attempts to link a theory of matter to thermodynamics in the form of a theory of gases were criticized and Maxwell initially read Clausius’ response to his critics. According to Clausius’ 1859 paper, the molecules of a gas roamed freely throughout its volume, the impacts between molecules were random and the volume of the molecules was much smaller than that of the gas. Clausius also introduced the ideas of the mean free path and the average velocity of the molecules. However, he did not investigate how to calculate these averages. He also related the pressure of the gas to the total kinetic energy of the molecules. On the basis of the ratio of the two specific heats, Clausius concluded that molecules were complex dynamical systems (Clausius 1859).

Maxwell took these papers, transformed the argument and extended it into a ‘conceptually self-contained predictive’ theory. In his first kinetic theory paper of 1860, Maxwell constructed a velocity distribution function and introduced probability arguments into physics. The focus of this paper was the deduction of expressions for the transport properties of gases, viscosity, diffusion and the conductivity of heat. From these he obtained the mean free paths of the molecules from which he estimated their size. He paid particular attention to viscosity because, theoretically, its value was independent of pressure (Garber *et al*. 1986). He set, with his wife, to investigate this, which he confirmed, and also showed that viscosity would vary as the square root of the temperature, as predicted by his kinetic theory, thus making the foundation of kinetic theory more plausible. In 1866 he was awarded the Royal Society's Bakerian Prize for these elegant experiments (Maxwell 1866). In addition, he could link the viscosity of a gas to the mean free path of the molecules and from thence estimate their size. His experiments on the pressure behaviour of gaseous viscosity also bolstered the plausibility of kinetic theory. His further experimental work on the temperature behaviour of the other transport properties of gases allowed him to discard the elastic sphere model of the molecule. In 1867, Maxwell established a far more general and defensible theory of gases based on molecules that were centres of force. His experiments on the temperature behaviour of viscosity led him to assume that this force was −1/*r*^{5} (Maxwell 1867).

In the middle of this work on kinetic theory, Maxwell returned to his paper on Saturn's rings. These notes were probably written between 1860 and 1866 because he used the methods of his 1860 kinetic theory paper (Brush *et al*. 1983, pp. 169–194). He ran into two problems. The first one lay in restricting the molecules to motions in a thin ring held in orbit by gravitation and subject to inelastic collisions. If the collisions were elastic, the particles would disperse into a cloud. Neither could he establish equations for the velocity distribution function that he could integrate. In a second attempt, he changed variables, but this approach also was inconclusive. As far as we know, he never returned to the problem. These notes remain as a bundle, some of them written as parts of partial papers, others less developed. Saturn's rings did not lead to kinetic theory, but quite the reverse. Yet, kinetic theory had not overcome the limitations of current theories of Saturn's rings (Brush *et al*. 1983).

## 5. Molecules

With his kinetic theory Maxwell had established a more defensible theory of matter. In Britain, even in the 1860s, some chemists were still arguing over the necessity for bothering with the ultimate structure of matter (Brock & Knight 1965). Until Maxwell's experiments on gases, there was no direct link between the gross properties of matter and its ultimate molecular constitution. Maxwell changed the grounds for the existence of molecules. In 1870, Maxwell addressed the British Association as president of Section A on the molecular aspects of his gas theory. After 5 years he presented the evidence for kinetic and molecular theories to the Chemical Society in London (Maxwell 1870, 1875*a*). For Maxwell and his colleagues, molecules became dynamical systems of atoms and the latter were stable and indivisible. In the 1870s Maxwell continued to publish on aspects of gas theory, especially that of Ludwig Boltzmann, most often focusing on its insights, and problems, in constructing molecular models. In his first gas theory paper, Boltzmann linked the ratio of the specific heats of gases to the number of atoms in a molecule of the gas (Boltzmann 1867).

In this era, the importance of gas theory was in the insights it gave into the structure of matter. However, kinetic theory faced serious problems in just this area. It could not explain the specific heats of gases. Clausius had assumed, and Maxwell had used, known conservation laws to argue that in the final state of a gas the average translational energy of the molecules is the same along each axis and equal to the average rotational energy of the molecule about each of the three principal axes of the molecule (Maxwell 1860). Maxwell was already sceptical of his sphere model of the molecule and this scepticism grew deeper when his calculated values of the ratio of specific heats based on this equipartition theorem did not agree with known experimental ones. However, his centre of force molecule led to the same dilemma that Maxwell had tried to explain away. He claimed that the ratio of specific heats was ‘merely to determine the value of a constant in the dynamical theory for agreement between theory and experiment’ (Maxwell 1867). What Maxwell missed was that the ratio of specific heats depended on his energy distribution function. The equipartition theorem depended on a distribution developed by assuming that the interactions of the molecules were the equivalent of elastic collisions. In not discerning the problem with the equipartition theorem, he missed its deeper implications. Later, Maxwell began to see the discrepancy between the theoretical and experimental specific heats as critical for the dynamical foundations of physics. He returned to this issue throughout the 1870s (Maxwell 1870, 1875*b*, 1877, 1879). However, it was only in 1900 that Lord Rayleigh revealed the destructive simplicity of the equipartition theorem (Rayleigh 1900).

The second problem was the complexity of the spectra of gases, research on which was coming to fruition in the 1860s. The abundance of lines from spectroscopic data indicated many more types of internal, atomic motions within molecules than allowed by kinetic theory. If there were only the few internal motions allowed by gas theory, how could these structures produce the myriad lines of atomic spectra? Maxwell tried to resolve these discrepancies by using various mechanical models for molecules but could see through the limitations of all of them (Maxwell 1871, 1875*b*). Yet Maxwell developed a qualitative theory of spectra. The sharp spectral lines came from the resonant vibrations of molecules excited during their mutual collisions. He used the analogy of the resonances of bells in a belfry, whose structures were hidden, but set in motion by pulling on the bell ropes. He was also led to the conclusion that all atoms across the universe were the same. This in turn implied that a universal constant existed which determined the frequency of their vibrations.

When Maxwell died in 1879, the issues stemming from the equipartition theorem and spectra were still far from resolved. Resolution required a new foundation for the theory of matter, quantum theory. Theoretical work in kinetic theory fostered a small experimental industry in the measurement of gaseous, transport coefficients throughout the rest of the nineteenth century. Yet, the discrepancies between theory and experiments were recognized and kinetic theory remained problematic. Lord Kelvin tried to undermine the foundations of kinetic theory through a series of ‘Test Cases’ published over a period of 20 years. The equipartition theorem became one of his ‘Clouds’ threatening a physics based in dynamics. His solution was to deny its results and hence keep the triumphs of nineteenth-century physics intact (Lord Kelvin 1902). To no avail, his examples were always countered. In addition Lord Rayleigh demonstrated that the problem lay not in kinetic theory but in the foundation upon which it was built, dynamics itself (Rayleigh 1900).

## 6. Concluding remarks

None of these disputes affected Maxwell's reputation. In the decade after his death, his theory of electromagnetism was deciphered through its adoption as a subject for the tripos examination. This meant that a small army of tutors, coaches and other members of the teaching staff at Cambridge delved deeply into Maxwell's *Treatise* (Warwick 2003). In the same era, Maxwell's work became central to the work of a far larger group of British, European and American engineers and physicists. The electrical industries expanded far beyond their beginnings in the telegraph industry of the 1840s and underwater cables of the 1860s. One notable example is Oliver Heaviside (1850–1925) who rewrote Maxwell's equations in vector form—a form more familiar to modern physicists. Much of Heaviside's work extended not only Maxwell's ideas but also incorporated work of his own. In 1887, the centrality of Maxwell's electromagnetic theory to late nineteenth-century physics was also assured with the publication of Heinrich Hertz's experimental demonstration of electromagnetic waves. By 1890, the foundation of Maxwell's fame rested on his theory of electricity and magnetism. His prominence in the development of molecular theory was almost forgotten. While very much a nineteenth-century physicist, Maxwell was able to see the limitations of molecular theory, the foundation of his reputation. The ability to publish on the limitations of the foundations on which one's professional credit rests is very rare in any profession, and one takes considerable risks in doing it. But Maxwell had shown that trait, of focusing on just those places where the limitations within a theory become apparent, even as a young, newly appointed professor, bringing in his baggage to the University of Aberdeen, investigating the Adams Prize essay question on Saturn's rings which began his career.

## Footnotes

One contribution of 20 to a Theme Issue ‘James Clerk Maxwell 150 years on’.

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