Geography made northern England the engine room of the industrial revolution. First came the mill towns, with clattering looms driven by fast-running water. In the bucolic south, placid rivers meandered across the rolling landscape, ambling between lush fields full of grazing cows. Farther north, torrents rushed in narrow channels down the steep slopes of the Pennine hills, where sheep nibbled the thin grass. In valleys where a few powerful streams came together, the mill towns blossomed. Tiny agricultural settlements quickly became home to thousands of mill workers—men, women, and children working six days a week for wages that barely kept them alive. This was the birth of the urban industrial working class.

By the end of the 18th century, steam power had begun its long ascendancy. The first steam engines appeared around 1700, and simple steam-powered pumps with a rocking action were used in a few mines not long afterward. But it took almost a century of inventions and improvements for rotary engines and, later, locomotives to become practical. Only then did the vast factories of the 19th-century industrial revolution begin to appear. Fast-flowing water from mountain streams was no longer a requirement, but steam engines needed water in bulk

and coal in large quantities. Industry moved down from the hills and into the flatlands. Cities sprang up beside large rivers, where coal and raw materials could be shipped in and finished goods shipped out. Manchester, Birmingham, Leeds, and many more sprawling, smoke-belching factory cities settled on old farmlands. Ports such as Liverpool and Glasgow grew too, bringing in raw materials, especially cotton and sugar, from British possessions around the world.

The transition from fast-running water to steam as the source of motive power had a striking scientific parallel. The working of a water mill explains itself with little difficulty. Water falls, turns a wheel, then goes on its way. Ideally, no water is lost. Nor is it immediately obvious that anything is lost from the water. If you go a little way downstream from a mill, the water may flow just as fast as it did upstream. Engineers of the 18th century were well aware, following Newton, that a water wheel generated mechanical work. And they knew too, if only from the obvious visual appearance of the thing, that this mechanical work had its origin in the fast-moving stream. But it seemed, from casual inspection, to be an inexhaustible source.

When scientists first began to ponder the working of steam engines, the analogy with water wheels proved irresistible. Heat from a furnace turns water into steam. Steam in the cylinder pushes on a piston, producing mechanical effort. Then, with the piston at full stretch, the steam (cooled because of its expansion) is vented into the air to allow the piston to slide back. From the hot furnace to the spent steam, a quantity of heat, it appeared, fell from a high temperature to a low temperature. Heat flowed, in some ill-defined sense, through a steam engine, yielding mechanical effort as it did so. It was not obvious that any heat was “used up” in the process. Indeed, the nature of heat being so enigmatic, it was impossible to say what using up heat might mean. Where would it go? What would it become? Far more reasonable to suppose that the quantity of heat stayed the same. Only its quality—its temperature—underwent any evident change.

***

During his 1845 stay in Paris, investigating with Regnault the properties of steam, William Thomson came across an intriguing paper writ-

ten in 1834 by Emile Clapeyron. Clapeyron gave a quantitative analysis relating the amount of work produced by a steam engine to the quantity of heat passing through it. More intriguing still, Clapeyron explained that his analysis was not original but rather was his attempt to put into tight mathematical form a prosy discussion he had found in a pamphlet written 10 years earlier, in 1824, by Sadi Carnot, and titled Réflexions sur la puissance motrice du feu et sur les machines propres à développer cette puissance (Reflections on the motive power of heat and on machines capable of developing that power, usually known by just the first half of the title). Carnot’s work, contemporary with Fourier’s great book, was utterly new to Thomson—and, as he quickly discovered, apparently unknown to the whole of Paris. The library at the Collège de France did not have it. Thomson went to bookstores and to the used-book stalls along the banks of the Seine asking, in his Scots-accented French, after Carnot. “Caino?” the merchants would respond, “Je ne connais pas cet auteur.” When he managed to convey the name correctly he was shown works on military engineering by Lazare Carnot and on social matters by Hippolyte Carnot. Of Sadi Carnot he found not a trace.

Lazare was Sadi’s father, Hippolyte his younger brother. Lazare Carnot was a man of some technical knowledge, who had applied his talents to military matters and then to politics, in which he showed impressive agility. He served with Robespierre on the Committee of Public Safety in the revolutionary 1790s and after a period of exile had come back as a minister under Napoleon in the early 1800s and again during the “hundred days” of 1815, when Napoleon returned from Elba only to be defeated at Waterloo and sent into more distant seclusion on St. Helena. Then finally Lazare’s ingenuity ran out. He went into exile in Prussia and died in 1823. His son Sadi, born in 1796, haphazardly educated, growing up in the shadow of political reverses, and with his father only intermittently present, became (according to his brother) a sullen and mistrustful man. After a brief military career, he hung about Paris in the occasional company of engineers and technical men but made little acquaintance with the world of science. Then in 1824 he produced one of the most profound and original scientific works of his or any era.

In pondering what went on inside a steam engine, Carnot’s acute insight was to think in terms of a repeating cycle. For the piston to begin

a new stroke, it had to come back to its starting position. Ideally, Carnot imagined, the engine would also start each cycle with precisely the same conditions of temperature and pressure in the cylinder. Each cycle was then identical and independent.

The conceptual obstacle to analyzing an engine was that complicated and interacting processes of heat transfer and work production occurred in what seemed to be a hopelessly entangled way. Carnot untangled the problem by idealizing his cycle so as to isolate the roles of heat and work. He imagined a furnace maintained at some constant high temperature, while the spent steam was expelled from the piston at some constant low temperature. He constructed a four-part cycle. First, the cylinder was charged with steam at the high temperature; second, the piston was released so that the charge of steam produced an amount of work, cooling and expanding as it did; third, spent steam was discharged at the low temperature; finally, the piston returned to the starting position.

Details aside, the trick here was that heat entered only at the high temperature and left only at the low temperature, in parts of the cycle where no work was being produced. During the work phase, on the other hand, no heat was moving in or out. By isolating the operations of an engine into distinct steps of the cycle, Carnot was able to compare heat going in to work produced and thus ponder the engine’s efficiency—the amount of motive power obtained from a given quantity of heat, or rather, as he explicitly said, a given quantity of the supposed heat-fluid called caloric. He declared firmly that “the production of motive power in a steam engine is due not to an actual consumption of caloric but to its passage from a hot body to a cold one” (Carnot’s italics). This is the mill wheel analogy. Heat passes through an engine, going from the high to the low temperature, but is not used up.

This sounds all very well, but Carnot’s cycle is by design highly idealized. How can it help in understanding the working and efficiency of a real engine—an actual machine with leaky gaskets, squeaking pistons, sticky valves, and all manner of other imperfections?

Here was where Carnot’s ingenuity blossomed into true originality. Because of the way he constructed it, his cycle was reversible. That is, if a certain amount of caloric moving from high to low temperature pro-

duced, via his ideal engine, a certain amount of work, then that same quantity of work, applied so as to run the engine backward, would push precisely the same amount of caloric “uphill,” from the lower to the higher temperature. Now imagine, he said, a hypothetical engine that could produce a greater amount of work from the same transfer of caloric. He could use part of the work from such an engine to run his cycle backward and push all the caloric back to the higher temperature—and he would have some work left over. This would mean he could create work by shunting a finite amount of caloric from high temperature to low and back again, ad infinitum. He declared roundly: “This would be not only perpetual motion, but an unlimited creation of motive power without consumption either of caloric or of any other agent whatever. Such a creation is entirely contrary to ideas now accepted, to the laws of mechanics and of sound physics. It is inadmissible.” Therefore, he concluded, the ideal cycle he had devised was the most efficient possible means to create work from transfer of caloric. No other engine could do better.

He offered one final extension of his reasoning. Engines did not have to run on steam. A Scotsman, the Reverend James Stirling, had invented an engine in which heated air pushed a piston. Now imagine, Carnot said, an ideal cycle running on air coupled to another cycle running on steam, but in reverse. If the efficiency of the air engine was greater than the efficiency of the steam engine, it would be possible as before to move heat uphill without a net expenditure of work. Therefore the efficiency of the two engines had to be the same.

To sum up, Carnot had constructed an idealized engine cycle with an efficiency that could not be surpassed by any other engine. It therefore represented a theoretical maximum. Moreover, this efficiency must be the same regardless of the engine’s working substance, so it was a universal theoretical maximum. Carnot stated with great emphasis the connection between maximum efficiency and reversibility. If any heat was lost in the running of the engine, whether from the escape of steam or from conduction through metal parts, it would not be restored if one ran the engine backward. The efficiency must then be smaller. Although Carnot concluded his analysis by using what knowledge he could find of the properties of steam to estimate actual work production by typical en-

gines, he could not establish a completely general formula. He did demonstrate, though, that the efficiency could depend only on the upper and lower temperatures between which the engine cycled.

Carnot’s essay of 1824, starting from next to nothing, created in a single burst of originality the foundations of a new science relating heat and work. It was utterly without precedent and dense with implications. Published privately in an edition of 600 copies, the Réflexions sank immediately into obscurity. Hardly anyone read it. Few who did understood it. And certainly it made no impact among French scientists.

For this Carnot has to take some of the blame. He argued in words, not mathematics, yet at the end provided tables of numbers representing the work that could be got out of an ideal steam engine working at certain temperatures, using a certain amount of coal, and so on. His style of exposition was clear and emphatic in places, cryptic and obscure at some crucial and delicate points. Carnot was an early exponent of scientific writing according to the principle “say what you’re going to say, say it, then say what you just said.” To mathematical scientists of the French school accustomed to tight algebraic exposition, his work came across as an exercise in rhetoric, with pages of numerical results arriving seemingly out of nowhere. To engineers occupied with making their furnaces and cylinders as heat tight as possible, and worrying about seals and friction and lever arms, Carnot’s abstract pronouncements about an imaginary kind of ideal cycle seemed spectacularly beside the point.

Even when Clapeyron, 10 years later, simplified some of Carnot’s arguments, left out some of his shakier reasoning, and wrote down a simple mathematical statement of the main result, he failed to attract much interest. The prospective audience—natural philosophers with mathematical inclinations yet interested in working out the theory of an industrial machine—barely existed. William Thomson, however, was the perfect reader. Heat had fascinated him since he encountered Fourier. He had embraced the strategy of tying mathematical reasoning to empirical knowledge rather than abstract principles. On top of that, he had grown up in an industrial city and, with his brother James, had made toy steam engines and other machines as a child.

He failed to find a copy of Carnot’s treatise in Paris, but he studied Clapeyron’s paper and, during the Thomson family’s summer outing at

Knock Castle on the Ayrshire coast, he told his brother James about it. Some months later James found a copy for himself and wrote to William: “The preliminary part, of wh. you told me the substance at Knock, is I think a very beautiful piece of reasoning, and of course is perfectly satisfactory.”

James Thomson, the professor’s oldest son, would have been celebrated as an extraordinarily bright young man had he not been blessed with an even more extraordinarily bright younger brother. He began attending his father’s lectures at the age of 10, but 8-year-old William already outshone him. Almost every year William won first prize in the class, and James came second. James was perhaps a little slower than his brother, but the more telling difference was that he tended (rather like his father) to be cautious and circumspect. Sometimes he seemed more profound or rigorous; at other times he could seem merely pedantic and unimaginative.

Three sketches, from the early, middle, and late parts of his life, furnish a consistent portrait of James Thomson. John Nichol, son of the astronomy professor, recalled: “Of the sons I liked James the best. He was crotchety and apt to be sulky with those who would not enter into his crotchets; here, as far as I know, his faults end.” Talking of his later career Nichol added: “I believe some of his inventions were excellent, but there was always some practical obstacle which prevented their bringing to the inventor either the fame or fortune they merited. James was an idealist in his way.”

In the 1860s the German scientist Hermann von Helmholtz visited William Thomson in Glasgow and while there met James. He wrote to his wife that James “is a level-headed fellow, full of good ideas, but cares for nothing except engineering, and talks about it ceaselessly all day and all night, so that nothing else can be got in when he is present. It is really comic to see how both brothers talk at one another, and neither listens, and each holds forth about quite different matters. But the engineer is the most stubborn, and generally gets through with his subject.”

When the brothers were old men, the engineer J. A. Ewing had much the same impression: “It was also, sometimes, difficult not to be impatient; for James, great as was his insight, seemed wanting in some sort of mental perspective, and had very little sense of time. There was never a

flaw in his logic; it was devastatingly thorough and would tolerate no admission of even the most obvious preliminaries. Occasionally one listened to his argument as the wedding guest listened to the tale of the ancient mariner, wondering not so much when it would end as when it would really begin.”

After their youthful studies in Glasgow, William did not take his degree, otherwise he could not have entered Cambridge as an undergraduate. James took his B.A. and went into a series of apprenticeships at engineering companies. There was no undergraduate engineering school to attend in those days. Technical men and inventors learned their craft on the job, sometimes picking up analytical skills along the way, sometimes not. In 1842 James started at Horseley’s, a company in Walsall, close to Birmingham in the smoky industrial heartland of England. While William studied, rowed, walked, and played music among the elegant buildings and sumptuous gardens of Cambridge, James wrote to him of a harsher reality: “I have a good many warnings about taking care of my fingers among the machinery. Mr Bell’s son has got 3 off his right hand and another of the pupils has just returned from London where he went to get his hand taken care of after having taken off one finger and destroyed another.” Another letter (from his lodgings, delightfully named Mrs. Grim’s) speaks of a boiler blowing up and killing a man. James helped draw up blueprints for iron bridges, which cost tens of thousands of pounds each; errors cost money, a factor that may have both suited and reinforced his natural inclination to extreme carefulness. But later letters complain of a lack of work coming into Horseley’s (as well as a lack of letters coming from William), and by early summer of 1843 the company had failed. James returned home to Glasgow.

The following year he went down to London, with no great enthusiasm, and found after only a couple of months that the company he had become attached to was about to be sold. By the end of the year he was in Manchester, in a pleasant part of town, he reported, not at all smoky except when the wind blew wrong. As William approached his final exams, James wrote to contrast his situation: “I wish my apprenticeship was as nearly done as yours, but even when it is done, I fear I shall have no such comfortable berth to step into as that which is probably waiting for you.”

Then ill health brought him down. He had often suffered through colds and infections, and on one of his walking trips in Europe he had damaged a knee that never fully healed. Early in 1845, fatigued and listless, he was diagnosed with a weak heart and returned to the family home in Glasgow where he received medical attention typical of the era. He had a “blister over my heart w^h kept me in bed for a fortnight” and afterward had “a silk cord put through my skin with the ends left out so as to cause a permanent running.” Regularly he received an “infusion of digitalis” to bring down the pulse, and he was instructed to take “no animal food or spirits of any kind.” A blister over the heart, it should be explained, was not an ailment but a treatment. A blister might be induced by burning or by the application of a poultice containing the dried bodies of cantharides, also known as Spanish fly or blister beetle. The irritation and subsequent infection were meant to scold the heart into working harder. A couple of years later Elizabeth was subjected to the same doctoring, and James wrote to William to say how he knew from experience that it was “really a most painful and distressing thing.”

Secluded in the Thomson home in Glasgow and unable to enter on any more apprenticeships, James pursued his theoretical investigations of engineering matters. While he and William were puzzling over Clapeyron, still unable to find Carnot’s original essay, their project of understanding the scientific principles of steam power came sharply up against a new and seemingly contradictory piece of information. In the summer of 1847, William went to Oxford for the annual meeting of the British Association for the Advancement of Science. This organization, founded in 1831 by a group of young reformers exasperated by the fuddy-duddies who ran the venerable Royal Society, had already established its annual meeting as the prime venue for announcing new results, sounding out one’s colleagues, and keeping abreast of the activities of scientific men across Great Britain. The BA attracted amateurs, gentlemen, engineers, and academics in equal measure, in contrast to the Royal Society, which had degenerated in the early 19th century into more of a London club for aristocratic dilettantes than a scientific organization. Facing pressure from the BA and other new scientific groups, the Royal Society had by midcentury largely regained its former reputation.

At the Oxford BA meeting, Thomson met James Prescott Joule, an

example of the new scientific man emerging from the industrial revolution. Born into a successful Manchester brewing family,¹ Joule had studied for a while with the great chemist John Dalton, a Mancunian who discovered strict numerical laws of proportions in chemical reactions and had gone on to propose the existence of atoms. Thereafter Joule largely educated himself in science and engineering, and had the resources to support a substantial laboratory in his own house. He was an active member of the Manchester Literary and Philosophical Society, established in 1781 as an intellectual forum for the emerging middle classes of the industrial city. The Lit & Phil, as it was known, began publishing its own scientific journal in 1785. True to its mercantile origins, the society held to a firm belief in the practicality of science. A visiting German scientist, Carl Jacobi, recalled speaking to the members of the Lit & Phil in 1842, when he “had the courage to say that it is the honour of science to be of no use, which provoked a powerful shaking of heads.”

While working in the family brewery, James Joule began to do scientific experiments. His first project, perhaps motivated in part by the rapidly growing abundance of noisy, smoky steam engines in Manchester, was an investigation of electric motors (invented by Faraday in 1821) as not only a cleaner alternative but potentially a more efficient and therefore cheaper one. It seemed not impossible at the time that electromagnetic motive power might be limitless. As with the analogy to water power, it appeared that a magnetic field, appropriately arranged, could cause an armature to rotate without itself being affected. This, Joule soon discovered, was not the case. He found that electricity passing down a wire creates heat in proportion to the square of the current. He found that as electric motors were made bigger, their coils somehow developed a resistance to the applied magnetic field that was trying to turn them. This was mysterious, but a general lesson urged itself on Joule’s mind. In modern idiom, you can’t get something for nothing. He observed that a current passing through an electromagnet will generate a certain amount

of heat; he then noted that if the same magnet was part of a motor, the amount of heat generated was reduced according to the amount of work the motor did.

Joule became a stickler for measuring things accurately and convinced himself that if one effect of an electric current—heat, mechanical power, a magnetic force—were somehow reduced, whatever was lost had to show up somewhere else, in some other form and, crucially important, in an equivalent amount. These studies led him to experiment on, and measure, the conversion of mechanical energy into heat. He arranged for a falling weight to turn a magnet and measured the current generated. He forced water through narrow pipes and measured the temperature increase. He used a known force to turn a paddle in an enclosed container of water and again looked for a temperature increase. Over a period of years he satisfied himself of a fundamental principle: A certain quantity of mechanical work, when efficiently and completely transformed, always created an equivalent amount of heat. This conversion factor he named the mechanical equivalent of heat, and in the ungainly units of the day he concluded that a quantity between about 600 and 1,000 foot-pounds of mechanical effort was needed to heat one pound of water by one degree Fahrenheit.

Getting his results published proved difficult, the Proceedings of the Royal Society being especially resistant. In later years he joked that these rejections didn’t surprise him. “I could imagine,” he said, “these gentlemen in London sitting around a table and saying to each other: ‘What good can come out of a town where they dine in the middle of the day?’”² Joule had more luck with the Lit & Phil journal and the Philosophical Magazine, a London journal founded in 1798 specifically to promote science of a practical, empirical nature. Even so, Joule’s presentations of his findings to British Association meetings, in Cork in 1843 and in Cambridge in 1845 (which Thomson is known to have attended), attracted mainly indifference.

In 1847 in Oxford he presented his latest results, using what he re-

garded as his most trustworthy method. By turning a paddle wheel to heat water, Joule had concluded that it took a little over 780 foot-pounds of effort to heat one pound of water through one degree Fahrenheit. In the audience was the new Glasgow professor, William Thomson, who at this time was wholly persuaded of Carnot’s principle that heat passed through a steam engine unchanged in quantity, creating mechanical work as it went. A Carnot engine running in reverse, therefore, used mechanical work to move a quantity of heat from a low temperature to a higher one, but now here was this man Joule saying that he could use mechanical work to create heat. According to his own recollection 35 years later, Thomson “felt strongly impelled at first to rise and say that [Joule’s conclusion] must be wrong,” but “as I listened on and on, I saw that … Joule had certainly a great truth and a great discovery, and a most important measurement to bring forward. So instead of rising with my objection to the meeting, I waited till it was over, and said my say to Joule himself, at the end of the meeting.” At a reception that evening at the elegant Radcliffe Camera, the two spoke further. “I gained ideas which had never entered my mind before, and I thought too I suggested something worthy of Joule’s consideration when I told him of Carnot’s theory,” Thomson recalled.

Though skeptical, Thomson was impressed by Joule’s modest sincerity and earnestness and by the obvious care with which he had conducted his experiments. He didn’t know what to make of Joule’s findings, but he saw something new and of profound significance. “Joule is I am sure wrong in many of his ideas, but he seems to have discovered some facts of extreme importance, as for instance that heat is developed by the fric[tion] of fluids in motion,” he wrote to his father, telling him to tell James to look out for reports of Joule’s work. He quickly developed a sympathy with Joule, who was overjoyed to find someone taking him seriously—and someone who was, by reputation, the rising star of British science. A little later Thomson sent copies of Joule’s works to his brother, with the warning “I enclose Joule’s papers which will astonish you.” James, in his measured way, wrote back two weeks later with his interim verdict: “I certainly think [Joule] has fallen into blunders [but] some of his views have a slight tendency to unsettle the mind as to the accuracy of Clapeyron’s principles.”

Thomson’s first meeting with Joule had an odd postscript. From the end of July to early September, Thomson traveled to Paris and then to Switzerland, meeting scientists but mainly enjoying a walking holiday. Near Chamonix, in the French Alps, he had an unexpected encounter. As he recalled it 35 years later: “Whom should I meet walking up but Joule, with a long thermometer in his hand, and a carriage with a lady in it not far off. He told me he had been married since we parted at Oxford! and he was going to try for elevation of temperature in waterfalls. We trysted to meet a few days later at Martigny, and look at the Cascade de Sallanches, to see if it might answer. We found it too much broken into spray.”

Looking for a temperature increase from the top to the bottom of a waterfall was a more hopeful than plausible way of finding out the heat generated by motion. Thomson’s charming tale, recounted many years later, is a fine example of his capacity for embellishment. At the time of the meeting he gave his father a simpler account. “Before leaving the St Martin road, I met, walking, Mr Joule, with whom I had recently become acquainted at Oxford. When I saw him before, he had no idea of being in Switzerland (he had even wished me to make some experiments on the temperature of water in waterfalls) but since that time had been married, & was now on his wedding tour. His wife was in a car, coming up a hill. As we were going different ways, we had of course only a few minutes to speak.” In other words, Thomson had told Joule he was going to Switzerland, and Joule had asked about the feasibility of measuring the temperature of waterfalls. The long thermometer that Joule carried while his new bride waited patiently in the carriage seems to be pure invention. Thomson was fond of these occasional ornamentations. To his credit, his inventions are rarely for self-aggrandizement, just to make a good story.

***

Settling into his new life in Glasgow, Thomson continued to puzzle over the apparent contradiction between Carnot and Joule but for the moment could see no way forward. In the meantime he developed his lecture courses, pondered other scientific problems and, judging by scraps of evidence from his correspondence, enjoyed a social life. Ludwig Fischer, his old Cambridge rival, later companion, had recently come to Scotland as professor of mathematics at St. Andrews University. Having seen a

note from Thomson to another Cambridge friend, Fischer wrote: “I must say I am not at all satisfied with the ‘pious’ wish you express at the end concerning matrimony, having hoped that your attention might have been much engaged at Oxford by certain young ladies, on whom, I learn from good authority, you have made the most favorable impression.”

Earlier in the year J. B. Dykes, an undergraduate musical friend on his way to a career in the church, had responded to some sort of jokingly admonitory letter from Thomson: “Your most grave & sober counsel had, I rejoice to say, a most beneficial & salutary influence upon me, & made me there & then, on the spot, repent in dust & ashes for my sins of omission & commission, mentioned by you in your epistle & more especially that heinous sin of flirtation. I felt most keenly the force of your remarks, & that they were so very much to the point inasmuch as I felt convinced that they came from ‘a party’ who was quite conversant with the topics on wh: he wrote & who in his daily & nightly serenades & promenades & searches after ‘them’ would have himself experienced so lately those pleasant & touching little sensations which he so wisely & properly reprehended in me…. Now don’t you go for to flirt with any young women at Oxford remember ‘them’….”

And when he took up his Glasgow professorship, another friend dashed off this warning: “Mind you don’t get married before you are aware of it—you are in a very dangerous position now—all the prudent mammas in Glasgow will be asking you to tea—but take care!” following up two months with a rumor lacking any foundation: “There is a tremendous report afloat in Cambridge about you—viz—that you are supposed to be married. I hope you will authorize me immediately to contradict it.”

These fragments give a sense of young men adopting a bluff and jocular style to hide their unworldliness. But Thomson was clearly no dry academic. The young wife of a Glasgow friend recalled: “I was asked to go to balls to chaperone him. The ballroom was a dirty trades hall badly lighted and with second rate music. William always used to ask me to take him home at 12 o’clock, but he was generally unwilling to come so soon….”

Elizabeth and Anna had tried teaching their brothers to dance when they were young, though at the time William in particular “professed utter scorn.” James never danced, but William evidently found a taste for

it, or for the benefits it brought, and Elizabeth suspected he took private lessons as a young Glasgow bachelor, though he was careful to conceal it.

When he had returned to Glasgow in 1846, the Thomson family was intact except for Anna, who was married and living in Belfast. Elizabeth had married the Reverend David King in 1843, and they lived elsewhere in the city. The rest still lived with their father, looked after by their Aunt Agnes Gall. The two younger Thomson sons, on whose education their father had lavished less personal attention, were adequate but not outstanding scholars. John, a lively and amusing youngster, at first went into the business world, learning the ropes in a commercial warehouse in Glasgow. But in May 1844 he was pleased to write to William in Cambridge to say that he had given up his job—“regular drudgery” he called it—and planned to study medicine. He would rather be happy than make money, he explained. He did well, winning the second prize in medical studies at the end of the 1846-1847 session, which was also William’s first session as professor. But studying to be a doctor was a perilous path. In April, while doing his rounds at the hospital, John caught a fever. Within a couple of days, at the age of 21, he was dead.

The following October Elizabeth, suffering from unspecified ill health, sailed to Jamaica to convalesce. She set off in a tearful farewell from the Glasgow docks, her father and siblings not at all sure they would see her again. That winter cholera struck Glasgow once more, and its victims this time included the 62-year-old and visibly aging Professor James Thomson. His end came quickly. He appeared weak but not overly unwell, then lapsed suddenly into a delirium, calling out urgently for his daughters and becoming calmer when he thought they were beside him. William described events in a letter to David King, Elizabeth’s husband. “He burst out rather faintly into a very incoherent set of expressions of numbers in all varieties of arithmetical denominations, hurrying rapidly from one to another, and giving the answer or saying ‘That’s right! Now, what is seven hundred and eighty-six inches equal to?’ and so on for several minutes.” Aunt Agnes wrote to Elizabeth of her father’s last moments. “Elizabeth! Elizabeth Thomson! Oh it is a dear name,” he called out. He died on January 12, 1849. Anna Bottomley came over from Belfast as soon as she could but too late to see her father alive.

Further departures followed. Robert Thomson, the youngest child,

had attained good health after surviving two surgeries to remove stones. In 1846 he joined the Scottish Amicable Insurance Office, starting on £20 a year—a tenth of the value of William’s Cambridge fellowship. William bought stock apparently on Robert’s advice, but was soon writing to his father asking for help on unloading it. In 1850, a year after his father died, Robert emigrated to Australia, where he married and had children. A letter from him to William survives, written in April 1885 on notepaper of the Colonial Mutual Life Assurance Society in Melbourne. It is a brief letter of introduction to William—Sir William Thomson by then—on behalf of a Melbourne colleague of Robert’s who was coming to England for some months. He never returned to Britain and died in 1905.

Recovering from her illness, Elizabeth returned to Glasgow. Meanwhile James, perhaps feeling overshadowed by his brother’s increasing reputation, moved to Belfast in 1851 and became a temporary professor of engineering at Queen’s College. He won permanent appointment in 1854. He and Anna were close, but in 1857 she died, at the age of 37, leaving a son, James Thomson Bottomley.

***

Carnot’s essay on motive power was not the only forgotten treatise that came to influence Thomson’s early career. In one of his undergraduate publications, Thomson had found a mathematical equivalence between the flow lines of heat, described by Fourier’s theory, and the geometry of electric lines of force, as proposed by Faraday. In this equivalence, contours of constant temperature corresponded to electrically charged surfaces. Thomson soon discovered he had not been as original as he first thought. In the Journal de Mathématique a few years earlier the Frenchman Michel Chasles had published related geometrical theorems, although he had not made the physical connection between heat and electricity. Thomson added a note to his paper in the Cambridge Mathematical Journal mentioning Chasles. But then he discovered a still earlier precedent. A brief citation in another paper suggested that both his and Chasles’s results had been anticipated in a work titled An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism, privately published in 1828 by George Green. A former Cambridge man, Green had died in 1841, and Thomson could find no trace of his obscure treatise.

According to another perhaps retrospectively enhanced anecdote, Thomson mentioned to Hopkins on the evening before he left Cambridge for Paris that he was intrigued by references to Green but hadn’t been able to find the Essay, whereupon Hopkins said he thought he had a copy. Going to Hopkins’s rooms, they found three copies, of which Thomson left with two, one for himself and one for Liouville. (An oddity of this tale is that Hopkins was surely familiar with Thomson’s published papers, apparently knew of Green’s Essay, but didn’t make the connection until Thomson brought it up. Either Hopkins, like everyone else, hadn’t read Green or he told Thomson about it earlier. But that would have spoiled the story.)

Green, Thomson now discovered, had established a whole range of mathematical theorems concerning the geometry of electric and magnetic forces and the distribution of charges and magnets. In Paris, word of Green got to another French mathematician, Charles Sturm, who had also published similar ideas. One evening an excited Sturm had come to Thomson’s lodgings on the Rue M. Le Prince, eager to see the fabled Essay. Riffling through the pages, he exclaimed “Ah! Voilà mon affaire!” when he caught sight of Green’s prior proof of his own theorem. Some years later Thomson arranged for the republication of Green’s work in a continental journal, along with his own explanatory essay, and certain theorems first associated with other names are now correctly known as Green’s.

At Liouville’s prompting, Thomson wrote a short proof of the equivalence of Faraday’s lines of force and the inverse square, action-at-a-distance picture preferred by the French. Now equipped with Green’s resurrected mathematics, he developed to a high degree of sophistication a new geometrical account of electric forces and charges. This work owed something to formal French rigor, to his own Cambridge training, as well as to Faraday’s vision. Most notable was his introduction of “images” in solving electrical problems. A conducting body, such as a metal sphere, carries electricity all over its surface when charged. The force between one such body and another, especially when their shapes are more complex, can be calculated in principle from the inverse square law, but only with difficulty. Between each point of one surface and each point of the other a force exists. The total force between the two extended bodies is

the sum of all these increments. Such a problem, an archetypal exercise in integral calculus, was a specialty of the French mathematicians, but for complex geometries the solution quickly becomes intractable.

Thomson proved that in terms of their electrical effects a charged body of some given geometry must be equivalent to a set of suitably placed points of electrical charge. Yet again Fourier’s treatment of heat flow provided the germ of the idea. A source of heat, or several, placed within some medium, will after a time lead to contours of temperature throughout the medium. Those contours bear a specific relationship to the heat sources that produced them. Likewise, Thomson showed, the conducting surface of an electrically charged body can be related to a set of charges with the appropriate arrangement—and calculating from a finite number of points is easier than dealing with an extended body of arbitrary shape.

Back in Cambridge, Thomson talked of his ideas at the British Association meeting there in June. The import, he explained, was that by taking Coulomb’s inverse square law of electrical attraction and repulsion, and applying the mathematical methods devised by Green, which he had partly rediscovered for himself, a number of Faraday’s assertions about the nature of electrical phenomena could be demonstrated. Faraday was at the meeting and spoke with Thomson, gratified that mathematical argument bore out his beliefs.

Despite this promising start, Faraday and Thomson corresponded only occasionally over the years. The two men’s mental powers, both acute, worked in utterly different ways. Thomson could never fully understand or even contemplate a proposition until he had given it precise mathematical form. Faraday, by contrast, constructed his physics entirely without the aid of mathematics, for the simple reason that he knew no mathematics.

No scientist, I believe, not even Newton or Einstein, had a greater power of pure imagination than Michael Faraday. The son of a blacksmith who had moved down from Yorkshire to London during economically troubled times, Faraday was born in 1791, the third of three children. His father had difficulty finding work and was often in poor health. The family lived in cramped conditions above a coach house in an area that today is on the fringe of London’s affluent West End. “My edu-

cation was of the most ordinary description, consisting of little more than the rudiments of reading, writing, and arithmetic at a common day-school. My hours out of school were passed at home and in the streets,” Faraday recalled. He left school at 13 and apprenticed to a bookseller and binder, George Riebau, who deserves recognition as one of the unsung heroes of scientific history. Apprenticeships were often little more than indentured servitude, but Riebau was a generous and large-spirited man. Faraday at first worked as an errand boy but soon began to learn bookbinding. He began to read the works he bound, and Riebau encouraged young Faraday to stay after hours and study whatever interested him. He read about electricity and chemistry in the Encyclopedia Britannica and with a few spare pennies bought old glass jars from a rag-and-bone shop to do his first experiments.

Industrialization and urbanization in the 19th century brought hordes of poor and uneducated young men into the growing cities. Philanthropists and social progressives, in their paternalistic but sincere Victorian way, founded evening schools and discussion societies to bring education and intellectual discourse to the working classes. The City Philosophical Society was one such institution. Faraday, joining it in early 1810, when he was 18 years old, participated nervously at first in discussions of history, philosophy, and science. Not unlike William Thomson’s father, Michael Faraday was single-minded in the task of self-improvement. Forming friendships with other young men, he sought to acquire good English and learn some French, and he put together a little chemistry laboratory to try out what he read.

Faraday was fanatical and orderly in taking notes and bound up his autodidactic writings in volumes that George Riebau showed off to some of his customers. One such regular, a Mr. Dance, was sufficiently impressed by the apprentice’s avid work that he gave Faraday tickets to hear lectures by the celebrated chemist Sir Humphrey Davy at the Royal Institution, near Piccadilly Circus and barely more than half a mile from the Faradays’ meager lodgings. Sir Humphrey was a dashing man and a thrilling speaker, apt to make the young ladies in his audience swoon. Faraday merely took notes and tried to perform at home the experiments Davy recounted.

When he was 21 Faraday’s apprenticeship came to an end. He was by

then too rapt by science to settle for the reliable but dull life of a bookbinder. He wrote to Davy for a job at the Royal Institution and got sympathy but no immediate help. A few weeks later, as luck would have it, Davy injured an eye in an experimental mishap and called on Faraday to assist him. Soon after that an assistant at the institution was thrown out for unruly behavior, and Faraday, in 1813, began working there, with accommodation provided and use of laboratory equipment in his spare time. In his long life he never worked anywhere else.

Six months later Faraday embarked on an 18-month grand tour of Europe with Davy and his new wife, the wealthy widow Mrs. Apreece, on the understanding he was to be Davy’s technical assistant. Lady Davy regarded him as a manservant. In the salons of the great cities of Europe, Davy parlayed his scientific talents into a kind of showmanship. In Paris he brought out his traveling chemistry kit and showed that a strange purple vapor was a new element, which he called iodine. In Florence he experimented on small diamonds that the Grand Duke of Tuscany sacrificed for science and proved that diamond was a form of pure carbon. Faraday met some of the great men of Europe, in between resisting Lady Davy’s instructions to haul luggage or shine shoes.

By the time they returned to England in 1815, Faraday had learned some chemistry and other science, but above all he had learned that salon life was not for him. He did not so much despise society as wish to live apart from it. Faraday’s family belonged to an exclusive and self-contained Protestant sect, the Sandemanians. They lived according to a strict and simple interpretation of biblical guidance written down in the middle 18th century by Robert Sandeman, who died in 1771 in Connecticut having failed to establish an American branch of his religion. The Sandemanians believed in salvation through faith and thus rejected as coarsely utilitarian the more usual Protestant idea of redemption through good works. They married among themselves, as Faraday did, marrying Sarah Bernard in 1821. Their social life was almost wholly among the Sandemanians. Faraday avoided as far as possible civic events and functions, even if he was the object of the honor, and in later years almost his only concession to the social graces was his annual attendance of the anniversary dinner of the Royal Institution. He had no students and rarely collaborated with others. He explained once: “I do not think I could

work in company, or think aloud, or explain my thoughts at the time. Sometimes I and my assistant have been in the laboratory for hours and days together, he preparing some lecture apparatus or cleaning up, and scarcely a word has passed between us.”

Central to the Sandemanians was a pious humility, a calm acceptance of the fallibility and imperfection of humanity. This attitude colored Faraday’s scientific work. “In all kinds of knowledge I perceive that my views are insufficient, and my judgement imperfect. In experiments I come to conclusions which, if partly right, are sure to be in part wrong; if I correct by other experiments, I advance a step, my old error is in part diminished, but is always left with a tinge of humanity, evidenced by its imperfection,” he wrote to his brother-in-law. Above all he turned away from worldly vanities, the false allure of reputation and public acclaim. Work was its own justification. The purpose of scientific investigation was to shed light, however feebly, on God’s design, and thus praise Him.

Even in the innocent days of the 19th century, such an attitude was hardly conducive to the promotion of a scientific career. Faraday resisted occasional attempts to draw him into professorial positions elsewhere and only intermittently attended meetings and conferences where he might explain his findings and opinions. After spending his early research years mainly on chemical work (notably he succeeded in liquefying chlorine), he moved into electrochemistry (reactions stimulated by the passage of electric currents through solutions) and thence into his pioneering and utterly original studies of electricity and magnetism.

Faraday was to an extent influenced, via Davy, by German philosophical views. Davy was close to the poet Coleridge, who had become a great proselytizer for Naturphilosophie after he spent 1798 in Germany. Kant, for reasons best left to philosophy, believed that the idea of points acting on each other through empty space was inadmissible. Instead he argued that forces pervading space were fundamental and that matter was in essence the manifestation of a resistance to force, rather than an entity in its own right.

However dubious these propositions, they stimulated Faraday to think of electric and magnetic effects as influences spreading throughout space, rather than as the summation of discrete forces between isolated objects. He objected vehemently to the idea that a force could act instan-

taneously across empty space. Instead, he thought electric and magnetic influences must propagate from one place to another, conveyed by some presumed medium—hence his lines of force, which he conceived almost as elastic links, carrying tension and perhaps inertia. Thus, Faraday viewed electricity and magnetism as live, conjoined creatures inhabiting space.

Qualitative though this picture was—as it had to be, since Faraday lacked the means to translate it into mathematical propositions—it enabled him to design and perform quantitative experiments. His most celebrated discovery was probably his demonstration of electromagnetic induction. It had been known since 1820 that a current passing along a wire would make an adjacent compass needle deflect. If a current could create a magnetic force, it seemed to Faraday and many others that the complementary effect—a magnet creating a current—ought to occur too. However, a permanent magnet placed beside a wire will do nothing.

In 1831, Faraday took an iron ring six inches across and wound it with coils of fine wire on opposite sides. Connecting a so-called galvanometer to one coil to detect any current, he touched the other coil to a battery and quickly disconnected it again. It was the pulse of current, not a steady flow, that made the galvanometer twitch one way when he connected the circuit, and the other way when he broke it again. Faraday imagined that when he passed a current through one of the coils, magnetic lines of force sprang away from it and cut through the other, generating a current. Once the magnetism was steady, the lines of force remained static, and no current appeared. But when he disconnected the circuit, the magnetic lines of force collapsed back again, generating an opposite current as they retreated through the secondary coil.³

Thus did Faraday’s conception of electromagnetism as a dynamic and extended phenomenon lead him to find a long-sought effect, where the old picture of static forces had borne no fruit. Later that year he showed that moving a plain bar magnet near a coil could also create a

current. The greater the number of turns in the coil, the more wires each magnetic line of force cut through as it moved and the greater the current produced. His conception of magnetism, qualitative though it was, yielded quantitative predictions.

William Thomson first encountered Faraday’s science in the early 1840s, when he was taking classes in Glasgow from David Thomson, substituting for the ailing Meikleham. It was then, Thomson claimed in his old age, that he was “inoculated with Faraday fire,” but in his Cambridge notebook from March 1843 we find him recording a long conversation with Gregory “in w^h Faraday and Daniell [another electrical scientist] got (abused)².”

Faraday devised his scientific theories in pictures, almost in cartoons, and though his experimental demonstrations were admirable, his thought processes must have seemed to young Thomson quaint at best. Where was the analytical proof? Where was the reduction of physical phenomena to quantities amenable to mathematical manipulation? Even now there is a tendency to praise Faraday with a tinge of condescension as a great but uneducated experimenter, as if he were some sort of idiot savant with an inexplicable knack for putting wires and magnets and batteries together in clever ways. Those who can think of physical theories only in mathematical terms evidently have trouble understanding a theoretician who did not work the same way. Faraday was no mathematician, but it was he more than anyone who originated the modern view of the electromagnetic field. He was a magnificent experimenter, but guiding his experiments was a powerful vision of electromagnetism. He had one of the great theoretical minds in physics.

Thomson slowly came to appreciate Faraday’s insight. Following their initial meeting and correspondence in 1845, Thomson began a series of papers under the title “Mathematical Theory of Electricity in Equilibrium,” in which he devised a general system for dealing with distributions of electric charges and the forces they produced. This was mathematics of a high order; it was also practical physics. For the small but growing number of practitioners who wished to make electrical calculations for the purpose of building machines and devices, Thomson’s methods were a boon.

Because this series of papers established relationships between

charges, considered as points in space, and the influences they produced, considered as effects spread throughout volumes or on surfaces, they made a start on capturing in mathematics Faraday’s picture of electricity as an extended and pervasive “tension” maintained somehow across space. But the point was arguable. Thomson’s mathematics could equally be seen as a comprehensive elaboration of the consequences of inverse square laws and action at a distance. In the first paper of the series, Thomson hedged his bets as to what his results meant. He preferred to think of them “merely as actual truths, without adopting any physical hypothesis, although the idea they naturally suggest is that of the propagation of some effect.”

The exchange of views between Faraday and Thomson, though slight, proved crucial. Thomson began to think, as Faraday did, of space as a medium supporting both electric and magnetic phenomena. How rapidly his thinking evolved is evident from a paper he wrote in 1847 and sent to Faraday. In it he showed, in a very preliminary way, how the properties of a physical medium could be connected to electric and magnetic influences.

Thomson imagined, in general terms, some kind of elastic solid, such as a lump of gelatin, which had both resilience and flexibility. Unlike a rigid solid, it would yield in response to an applied force; unlike a liquid, it would rebound to its original state on removal of the force. Thomson showed that from a purely mathematical standpoint, forces of electric attraction or repulsion had the same form as compression or tension in the medium. Magnetism was trickier. A compass needle deflected by a current passing along a wire, for example, might flick to the left above the wire but to the right below. The geometry of magnetic forces, Thomson showed, paralleled a mathematical description of a localized twist or rotation of the elastic medium (as if, loosely speaking, one held a lump of gelatin in one hand, stuck a fork in it with the other, and twisted the fork a little).

“What I have written is merely a sketch of the mathematical analogy,” he explained to Faraday. “I did not venture even to hint at the possibility of making it the foundation of a physical theory of the propagation of electric and magnetic forces.” Nonetheless a physical theory was what he had in mind, at least as a distant dream. He had abandoned

the old action-at-a-distance philosophy in which one simply posited forces between particles in truly empty space. The idea of space as an electromagnetic medium was pressing on him. Since the same medium, he hoped, would eventually be seen to carry both kinds of effects, there was the ultimate prospect of connecting electricity and magnetism by means of a single fundamental theory. This was to be the preoccupation of a lifetime, but for the moment Thomson contented himself with working out a couple of long accounts of the geometry and mathematics of magnetic forces, as he had done for electricity. Further study of the presumed medium supporting these phenomena would come later.

***

Ten days after the death of Professor James Thomson, Hugh Blackburn wrote to Thomson to say he intended to put his name up for the vacant position. Like many Cambridge-trained mathematicians (at least those who didn’t want to enter the church), Blackburn had gone down to London to take up a legal career, but unlike Archibald Smith, he was finding no satisfaction in it. Thomson, while offering encouragement to his old undergraduate friend, tried to interest his colleague George Gabriel Stokes in coming to Glasgow. Stokes, five years older than Thomson, had been senior wrangler in 1841 and then became a fellow at Pembroke College. Like Thomson, he applied his mathematical knowledge to questions of physics and achieved important results in optics and fluid mechanics. His Cambridge career was not yet secure, however, and the prospect of a lifetime appointment alongside his friend Thomson held many attractions. But he ran up against Glasgow rules. All professors had to sign the Westminster confession, by which they declared their allegiance to the Presbyterian Church of Scotland. This was meant to guarantee that faculty members would take no part in the kinds of religious turbulence that had disrupted Scottish life and politics since the days of the Covenanters.

With his detestation of religious prejudice and sectarianism, Thomson’s father had campaigned against the religious test as a needless holdover from unenlightened times. It survived, largely because signing the confession was seen by younger men as a piece of meaningless bureaucracy. Stokes, however, was the son of an Anglican minister in County

Sligo, Ireland, and his three brothers all became churchmen. He was moreover a punctilious man. He arranged to have testimonials sent to the Glasgow faculty but immediately regretted doing so. A month after James Thomson’s death he wrote to William to explain that after consulting his older brother he decided he could not go through with the application. The “straightforward course is, to decline to take [the religious test] unless I am prepared to become a thorough Presbyterian, which certainly I do not mean to become…. It was all along a very doubtful question with me whether I could sign the test in a lax sense.”

Laxity was fine with Thomson, however. Whereas his father had argued for the abolition of the test, as a matter of principle, Thomson’s solution was to let the thing slide. He himself, he explained to Stokes, regularly attended the Episcopal Church in Glasgow (the Scottish counterpart of the Church of England) and went to the Established Church (of Scotland) no more than “once or twice or three times in the course of a session.” Neither he nor his colleagues found anything objectionable in this. He told Stokes that “the amount of conformity to the Established Church which a conscientious observance by one in your position of the obligations imposed by the tests, would really be in no way inconvenient, or repugnant to your feelings.”

He added: “It will be a very serious blow to the interest of this University if an honest member of the Church of England should never be able to be a candidate for any situation or office connected with it, however valuable an acquisition he might be; on account of an act of Parliament framed at a period of great political & ecclesiastical excitement; and allowed to continue unmodified in these settled times, merely because the modifications that those who have the interests of the University most at heart would be inclined to have made, are such that only those parts of the Act which are at present practically inoperative, would be abolished.” No doubt this was an accurate assessment. The rule was still on the books, but everyone agreed to look the other way, except perhaps for some of the older and more conservative men, whose opinions hardly counted anyway.

Thomson was a conventionally religious man all his life and believed firmly that the rational working of the universe and the ability of science to describe it were signs of God’s immanent power. But for niceties of

doctrine and points of observance he had no patience. Long ago, in the summer of 1834 or 1835, the vacationing Thomson family had gone to services at a local parish church where there happened to be a revival meeting. Their father was not there, and Elizabeth had charge of the boisterous youngsters. As the service proceeded and the preacher became more animated, members of the congregation began to moan and sway and throw their bibles in the air. This set William snickering, which set off the other children. The preacher, hearing the disturbance, interrupted his sermon, glared at the Thomson children, and pointed his finger at William. “Ye’ll no lach when ye’re in hell!” he thundered, at which William and the rest collapsed into a helpless fit of the giggles, leaving Elizabeth, red faced, to hustle the young heathens from the church.

As an adult William learned to maintain his decorum, but he seems to have regarded church going as a necessary formality, bearing little relationship to his thoughts on God and the nature of the physical world. Stokes, by contrast, took these things seriously and would not go along with Thomson’s plan to sign the Glasgow religious test with his eyes closed, so to speak. In April 1849 the open mathematics chair went to Hugh Blackburn. Later that year Stokes became Lucasian Professor (Isaac Newton’s old position) and remained at Cambridge the rest of his life. Blackburn performed adequately as a teacher of mathematics but made no original contributions to his discipline. Stokes wrote voluminously on mathematical physics, ended up with a theorem,⁴ an equation in fluid mechanics, and some optical phenomena named after him, served for a long time as secretary of the Royal Society, oversaw in minute detail the production of the society’s Proceedings, and acted, through his indefatigable correspondence, as a guide and mentor to numerous mathematical physicists in Britain, Thomson included. That this career was lost to Glasgow University because of an antiquated rule might have been, in Professor James Thomson’s hand, an additional spur to long overdue reform. To William Thomson it was a matter of keen but passing regret.

He wrote to Stokes that “no case can prove the noxiousness of the [Glasgow] law … than the present one,” but he took the question no further.

By contrast, Thomson threw himself with great energy into his new duties as Glasgow professor. He composed an opening lecture for the incoming class, which he delivered with minor variations each year for many decades. Science, he explained, began with natural history, which was the close observation and classification of material phenomena, and rose to the level of natural philosophy, which was the attempt to understand and connect those phenomena by rational means, expressed ultimately in the language of mathematics. Mechanics was the most mature of sciences, while electricity and magnetism were approaching that pinnacle. He threw in snippets from Francis Bacon and talked of the practicality and applications of science. He added a dash of religion, to say how science aimed to illuminate God’s handiwork: “When I consider thy heavens, the work of thy fingers, the moon and the stars which thou hast ordained; What is man that thou art mindful of him, and the son of man that thou visitest him?” Science was to be undertaken in a spirit of humility and with a sense of wonder and beauty; nor should the ability of science to improve the lot of mankind be ignored.

This was the nearest Thomson ever came to a philosophical statement of purpose. He struggled to compose the lecture and was not pleased with his first delivery. “According to his own account, it was a total failure,” Elizabeth wrote to her husband. “I think he had been very nervous, and he read much too fast…. He is very much disheartened, poor fellow.” As a lecturer, Thomson tended to be more enthusiastic than orderly. He tried to keep to a plan but could never resist the digressions that rained in on his mind. At his best, to an appreciative and sophisticated audience, he could be thrilling, inventing profound and provocative science as he spoke. When he gave his introductory lecture he generally tried harder to stick to his script, and the struggle detracted from his fluency and intensity. James Clerk Maxwell, the young Scots physicist who began to make inroads into electromagnetism in the middle 1850s, said that Thomson’s annual introductory lecture never managed to fill the hour and that “the lecturer was greatly downhearted at its conclusion.”

With this weak essay at a grand purpose out of the way, however, Thomson moved into his regular scientific lectures with eagerness and delight. There he was never downhearted, always ready to share the joys of discovery and enlightenment with his pupils. But in his success at imparting information into lesser minds than his own, he got mixed reviews. Bright students liked his style. One recalled him as “an enthusiastic and inspiring teacher; he aroused and sustained the intelligent interest of his students…. No one could listen to him without being imbued with his spirit and being borne along the path he was travelling…. He was always in earnest, and when dealing with great problems spoke with the fervour of a missionary charged with a weighty message.” But another student, less overawed, more overwhelmed, said: “Explanation, it has to be confessed, was never his forte. He would say, ‘Look, see it and believe it.’” And from another: “Even in his introductory lectures Thomson soared to heights which made many of his class feel giddy and helpless.”

Though he was patient with slower students, he seemed to think they were being obtuse, not that they had genuine difficulty understanding. He would prompt a struggling pupil more and more minutely, in smaller, easier steps, and finally say with bafflement rather than exasperation: “Now, Mr. Macintosh, why could you not have said so at first; why will you have me drag the information from you sentence by sentence, clause by clause, nay word by word?”

Particularly distressing to Thomson was what he took to calling aphasia—the inexplicable but frequent phenomenon by which capable students were reduced to helpless, struggling silence by the most elementary of mathematical questions. These students, he said, “will not answer when questioned, even when the very words of the answer are put in their mouths, or when the answer is simply ‘yes’ or ‘no.’” Thomson read mathematics as easily as he read words and could not understand why others did not have the same facility.

As a professor of natural philosophy he deserves credit for one fundamental innovation, which was the teaching of practical science through student experimentation. With his brother James he had made mechanical toys, but not until his visit to Paris did he attempt any measurement or manipulation in a scientific laboratory. Neither at Glasgow nor at

Cambridge nor anywhere else in Great Britain was experimental science taught; instead, men such as Joule figured it out for themselves, with some advice from their elders, while in Cambridge Stokes and others began to take up laboratory work on their own initiative and using their own resources.

William Meikleham had not taught any experimental science and undertook no significant research either. When Thomson assessed his new professorial domain, he “found apparatus of a very old-fashioned kind. Much of it was more than a hundred years old, little of less than fifty years old, and most of it was of worm-eaten mahogany…. There was absolutely no provision of any kind for experimental investigation, still less idea, even, for anything like students’ practical work.” He credited Thomas Thomson, his former teacher and now faculty colleague at Glasgow, with having founded a laboratory for his chemistry students to work in. He wanted to work on problems in electromagnetism, and after setting up a laboratory he engaged his students to assist him. This became an essential part of his course in natural philosophy. He did not merely teach mathematical methods and explain what crucial information had emerged from experiments by others. He got the students to do the experiments themselves and to explore the subject in a practical way. It is hard to imagine now how physics could have been taught in a purely abstract way. But before 1846 it was, and Thomson was the first instructor in experimental physics.

To further both his teaching and his research, he embarked on a battle with the Glasgow faculty reminiscent of his purchase of the “funny” only a few years earlier. He applied for money to buy equipment, spent it, spent more, and argued with his colleagues until they paid up. He took over vacant rooms near his lecture hall, turning an old wine cellar into his first laboratory, and when a larger room became vacant after some administrative change, Thomson “instantly annexed it (it was very convenient, adjoining the old wine-cellar and below the apparatus room); and, as soon as it could conveniently be done, obtained the sanction of the Faculty for the annexation.” There were protests and exchanges of letters, as there had been with his father, but Thomson committed the fait accompli and got his way.

In both lecture room and laboratory, Thomson invariably slid from

an exposition of textbook material into wide-ranging discussions of unsolved problems currently on his mind. This might confuse his students; it could also enthrall them. Sometimes they tried out standard exercises in the use of laboratory instruments; sometimes they helped Thomson with his latest research project. In overseeing such enterprises the young professor “had none of the air or manner of a superior.” As a teacher he may have flown often above the heads of his students but “he was never dull, never trivial, never commonplace.” “What I liked best,” said another student, “was when he left us to follow or not as we could, and went on thinking aloud, as he sometimes did. His mind was full of fancies, brimming over with metaphors.”

***

With his father and brother John dead, both sisters married and, after 1851, brothers James in Belfast and Robert gone to Australia, William Thomson was alone in the old family home except for his aunt and housekeeper, Agnes Gall. He went to dances with Jemima Blackburn, Hugh’s wife, as chaperone. Flirtations of an indeterminate nature came and went. His old friend Ludwig Fischer, at St. Andrews, wrote to him early in 1850: “I suppose it is out of the question your coming next week. Else we mean to have a Bachelor’s ball on Thurs the 28th, and I might have to offer you a faint resemblance of what you enjoyed at Edinburgh. Of course the Ladies of Scotch Craig, I spoke to you of have been invited. But I doubt whether they would come; nor have I heard whether the beautiful Fanny will be there.”

Early in 1851 Thomson was elected fellow of the Royal Society, a few months before his 27th birthday. In that same year he twice proposed marriage to Sabina Smith, sister of Archibald.⁵ Although Archibald Smith had encouraged Thomson’s academic career, congratulated him warmly on his successes, and in the end made no serious move to compete for the Glasgow chair, he advised his sister that “I really do not think you would be suited to each other.” She duly turned William

down. A year later he tried again, for a third time. Again, at her brother’s urging though apparently against her own inclination, Sabina said no. William confessed to Sabina’s sister his “bitter bitter grief” and many years later Sabina wrote of her regret at not resisting Archibald’s influence: “It was the extremity of folly to think I c^d go on refusing a man, & yet have him at my disposal whenever I choose!” By this time, however, she had seen young William Thomson evolve, over the decades, into the wealthy and celebrated Lord Kelvin, a great figure in the land. This may have amplified Sabina’s remorse.

Thomson was on his own now. His father was no longer around to advise him or to plan his campaigns. In any case, James Thomson had detested old Mr. Smith and was no great admirer of Archibald, so perhaps would have advised William against this entanglement in the first place.

In July 1852, only three months after his final refusal by Sabina Smith, William became engaged to Margaret Crum, whom he had known since childhood. She was the daughter of Walter Crum, a Glasgow cotton merchant who was first cousin to William’s father. Margaret knew his sisters, and their letters to him while at Cambridge mention a number of visits by her but say little about her activities, interests, or personality. The betrothal was sudden. To Elizabeth William wrote of news which “I think will please you as much as it will surprise you.” He told Stokes that “sometime, probably early in September, I am going to be married, to a Miss Crum. I cannot describe her exactly to you, but I am sure that is unnecessary to ensure your good wishes at present, and when you come down to see us in Scotland, I am sure you will be glad to make her acquaintance.” Unfortunately no one else seems to have exactly described Margaret Crum either. She and William appear to have embarked on some kind of alliance rather than a romance. As Margaret explained to Elizabeth, “We have one interest in common that can never fail, and as I told Mrs. Gall, I feel that in William’s love for his sisters and her, lies my best security for the continuation of those feelings on which the happiness of my life must now depend.” William apparently set aside passion and deep feelings of the soul after he finished reading Evelina and Wilhelm Meister in Cambridge, and his disappointment over Sabina Smith stifled any resurgence.

William Thomson and Margaret Crum were married in Glasgow on September 15, 1852. He was 28 years old; she was 22. From a brief honeymoon in north Wales the following week, William sent this account of an afternoon with his new wife: “The day is somewhat dark and cold, and some people might say dreary, but it does not seem so at all to me. I scarcely think it does so to Margaret either, although she has just been saying to me that it is, and what is more, laying particular emphasis on the most dismal parts. Perhaps she is only joking, but whether or not, she looks cheerful, and has quite got rid of her cold. In fact, I do not think either of us are going to apply to Dr. Brown to undo what he did on Tuesday.”

One may best interpret this as a piece of dour Scots humor or as the effort of a young man trying to impress an older married sister with his newly acquired sophistication. Few other impressions of Margaret are to be found. Jemima Blackburn recounts how William was a great friend and frequent visitor to the Blackburn household, both before and after his marriage, but his wife makes no appearance in her recollections. Hermann von Helmholtz, encountering her for the first time, wrote of “a rather pretty woman, very charming and intellectual” and the novelist William Thackeray, having met the Thomsons in Glasgow through mutual friends, asked to “give my best regards to … Thomson please with his nice wife.” On the other hand, she was an amateur poet. The first lines of her poems (published privately after her death) display a grim consistency, and a selection of them can be arranged almost to form a poem themselves:

Margaret’s verses are doleful, monotonous, and self-absorbed. This was a woman at least half in love with easeful death. To be fair, she had grounds for misery. In May 1853, eight months after their marriage, the

Thomsons went on a Mediterranean tour, taking in Gibraltar, Malta, and Sicily. To William, long accustomed to hiking in the highlands, this was a mere jaunt, but Margaret wore herself out, came home weak and ill, and though the nature of ailment remains unclear she was an invalid for the rest of her life. Back in Scotland she stayed several weeks in Edinburgh for “surgical nursing.” Helmholtz, after meeting her, offered this explanation: Margaret was “in a wretched state. A year after her wedding she suffered an abdominal inflammation,⁶ and for two years has been in such a state that she can’t walk, stand or sit upright without pain, and can only lie on her back.” Whatever the cause, Margaret’s health was such that she composed a poem, “On Pain,” from which the following selection is ample:

Thomson became nursemaid to his wife. By next summer her health had not significantly improved, although she had occasional better days that William seized on as grounds for hope. He reported to Elizabeth that “she looks much better … but she has not at all advanced in walking power. I always carry her up and down stairs, and often from one room to another. A walk half round Miss Graham’s garden lately knocked her up for several days. But by avoiding all such exertion she keeps tolerably free from pain, and has much the appearance of good health. I take her a drive nearly every day and sometimes twice in a little pony carriage.”

The appearance of good health is the most Margaret Thomson ever subsequently attained and that only intermittently. It is easy for the modern reader to infer some variety of malingering on Margaret’s part, perhaps amplifying an underlying problem. William was clearly of so accommodating a nature that his wife had no great incentive to improve, especially if, as her poems suggest, she had developed a fond intimacy with chronic pain and morbid thoughts. But we should not forget that in the middle of the 19th century all kinds of internal disorders were undiagnosable and untreatable.

In the years thereafter Thomson applied himself diligently and uncomplainingly to the care of his invalid wife. She became a duty rather than a passion. Perhaps that suited him.

***

After three years of thinking mostly about electricity and magnetism, Thomson returned to his Paris discovery of the science of steam engines. Still he had not seen Carnot’s essay and knew it only through Clapeyron. But that was enough for him to turn one aspect of Carnot’s theory into an important realization, both theoretical and practical, about temperature. Hot and cold, of course, are direct physiological sensations, but temperature is a difficult concept to make quantitative. Around 1590, Galileo invented his “thermoscope,” in which the expansion of air with rising temperature provided a crude numerical scale. Early in the 18th century Gabriel Fahrenheit came up with recognizably modern thermometers that relied on the expansion of colored alcohol or mercury. There was no theory to speak of behind these instruments, only the empirical fact that gases and liquids tend to expand when they get hot. But these

early thermometers at least allowed measurements by different investigators to be recorded, compared, and calibrated against each other.

During the 18th century many experimenters studied the relationship between pressure and temperature for a given volume of gas. Their results came together in a simple law: pressure times volume rises in proportion to temperature. In particular, for a gas maintained at constant pressure, volume changes linearly with temperature, which meant that the expansion of any suitable gas would serve to make a thermometer and that all gas thermometers ought to yield the same temperature scale.

It was readily apparent, though, that this wasn’t quite true. The simple rule relating pressure, temperature, and volume became known as the ideal gas law, on the understanding that all actual gases departed from this ideal in ways small or large. Gas thermometers, therefore, gave slightly different temperature scales depending on which gas was used. One scale could always be calibrated against another, but since there was no independent way of measuring temperature, there was no way to say which temperature was most nearly correct—that is, which one corresponded most closely to the temperature implicit in the ideal gas law.

Also implicit in the ideal gas law, it appears, is that temperature cannot fall without limit. On cooling, the volume of any gas decreases, and since no physical object can have zero volume, let alone a negative volume, there would seem to be an absolute zero of temperature, below which it cannot fall further. As early as 1699, the Frenchman Guillaume Amontons estimated that this endpoint corresponded to a temperature of about −248° Celsius, and in 1847 Thomson’s friend Regnault came up with −272.75°C, very close to the modern value of −273.15°C.

But this is highly misleading. A volume of steam, for example, cooled below 100°C at normal atmospheric pressure, turns into water. Below that temperature the gas law no longer applies. Likewise, although air could be cooled to much lower temperatures without apparent change, no one in the late 18th or early 19th century imagined that the temperature could really be reduced down to −270°C or thereabouts without some substantial physical change intervening. Regnault and his contemporaries therefore regarded a numerical value such as −272.75°C as a calibration point for a gas-based temperature scale, and nothing more. Absolute zero as a physical concept did not yet exist.

In Regnault’s lab and elsewhere the air thermometer had become the de facto standard, and it could be calibrated with some consistency against mercury or alcohol thermometers. Still, no rational temperature scale—meaning a temperature that was defined quantitatively in terms of known physical laws and standards—had been devised. Thomson concluded that the vagaries of individual gases and their failure to live up to the ideal gas law prevented any gas-based thermometer from yielding an absolute temperature.

In 1848 Thomson wrote a short paper explaining how Carnot’s theory could solve the problem. Carnot had established that the maximum work any engine can produce from a known quantity of heat can depend only on the upper and lower temperatures between which the engine cycles, and that this efficiency is the same no matter what the working substance, whether steam, air, or some other gas. Thomson defined a temperature scale by asserting that a Carnot cycle operating through a one-degree interval always produced the same amount of work from a given quantity of heat. That is, a certain cycle would yield the same amount of work operating between 100° and 99° as it would operating between 99° and 98°, and so on.

Thomson’s noteworthy conceptual innovation here was to define a temperature in purely mechanical terms, independent of the properties of this or that gas or liquid. On the other hand, since building an ideal Carnot cycle was no more possible than finding an ideal gas, his temperature definition was at this point theoretical rather than practical.

By asserting that an ideal engine had the same efficiency at all temperatures, Thomson was able to say that “all degrees have the same [mechanical] value.” But this was no more than an assumption. Thomson could cite no evidence or argument for it.

A corollary of this assumption was that Thomson’s temperature scale had no zero. Drop a degree, get some work; drop another degree, get the same work again; and so on without limit. His temperature scale therefore went down to minus infinity. He knew that the temperature scale defined by an air thermometer came to a halt at −273°C, when volume went to zero, and so concluded that “infinite cold [on his scale] must correspond to a finite number of degrees of the air-thermometer below zero.”

Histories of science sometimes claim that Thomson’s 1848 paper established the existence of absolute zero as a physical concept. This is not true. He clearly regarded it as a virtue of his temperature scale that all degrees had “equal value” and that it went down to “infinite cold.” By contrast, he said, “the value of a degree … of the air-thermometer depends on the part of the scale in which it is taken.” In other words, the fact that gas temperature had a zero in the vicinity of −273°C he did not regard as physically meaningful but as a misleading consequence of the way it was defined.

In setting out these conclusions, however, Thomson admitted to one nagging anxiety. He still had not resolved the apparent contradiction between Carnot and Joule. He persisted with Carnot’s view that the production of mechanical work during a cycle came from the transmission of heat from a higher to a lower temperature. But in a footnote he referred to “Mr. Joule of Manchester,” who had unarguably converted work into a proportional amount of heat. Joule believed that the reverse must also happen, but Thomson declared that “the conversion of heat (or caloric) into mechanical effect is probably impossible, certainly undiscovered.” Rather helplessly he could only conclude that “much is involved in mystery with reference to these fundamental questions of Natural Philosophy.”

Just weeks after this paper appeared, Thomson acquired a copy of Carnot’s essay on motive power from Lewis Gordon, who had been appointed in 1840 the first professor of engineering at Glasgow. Thrilled to see at last the source of ideas he had been pondering so long, he immediately set to turn Carnot’s wordy discussion into a logical and mathematical exposition. He talked of his discoveries to colleagues, and in April 1848, J. D. Forbes, a professor at Edinburgh, wrote to Thomson urging him to write up his analysis for the Royal Society of Edinburgh. “As you have taken so much trouble about this Theory of Carnot’s,” Forbes wrote, “I think it would be reasonable to expect you to print a little notice of it for the benefit of people in general.” Excited though Thomson may have been, Forbes had to prod him again in November: “I write to remind you of your promise to give us an Abstract of the Motive Power of Heat for the R.S. When can we have it?”

Delivered in January 1849, Thomson’s “Account of Carnot’s Theory

of the Motive Power of Heat, with Numerical Results deduced from Regnault’s Experiments on Steam” played a pivotal historical role. He brought Carnot, ignored by the French scientists and unknown to the English, before a new audience and did it in a way his colleagues could grasp. A direct translation of Carnot into English might have had as little effect as his original publication in French. Thomson’s interpretation and amplification of Carnot not only rescued the Frenchman’s seminal work from a quarter century of obscurity, but showed how these new ideas could be expressed in the modern language of mathematical reasoning. Thomson here bestowed a new name on this area of study. “Thermodynamics” he called it—the dynamics of heat, the connection between heat and work.

Even as he was preparing his account of Carnot, Thomson’s grasp of this new science evolved in fits and starts. Writing to Forbes, he teasingly mentioned that he had thought up a trick for producing ice “ad libitum without the expenditure of mechanical effect.” A Carnot engine running in reverse moved heat from a lower to a higher temperature. It struck Thomson that if both temperatures were the same, then an ideal engine (with frictionless moving parts, no heat leakage, and so on) would still move heat from one side to the other but would consume no physical effort because there was no change of temperature. So he imagined an engine operating between two reservoirs of water, each at precisely 32°F. Extracting heat from one side would turn the water into ice but with no mechanical cost. Hence his note to Forbes.

But when William tried this out on James Thomson, his cautious and thoughtful brother immediately saw a difficulty. Water expands when it freezes, and the expansion could be made to push a piston and do work. If that were the case, William’s ice machine would apparently create mechanical effort out of nothing, which they both deemed unacceptable. James found the answer. He concluded that the melting point of ice must fall slightly when pressure is applied. Then any attempt to make the ice do work would unfreeze it, and it could not push a piston. Some months later William did experiments to check this prediction and found that James was exactly right, and had accurately calculated the magnitude of the effect. The result thrilled William. In his Glasgow lectures he always described this finding with delight, explaining that it was the first

quantitatively precise prediction to be derived from Carnot’s theory of engines.

The failed ice machine didn’t appear in his presentation to the Edinburgh Royal Society, but plenty of other puzzles remained. The disagreement between Carnot and Joule worried him still. He fully accepted by now Joule’s numerous demonstrations of the conversion of heat into work, but he continued to insist on Carnot’s principle that, as he put it, “the thermal agency by which mechanical effect may be obtained, is the transference of heat from one body to another at lower temperature.” Heat is not consumed. Somewhat desperately, he claimed that this principle had “never been questioned by practical engineers,” although he would have been hard pressed to find a practical engineer who even understood the question, let alone had an answer.

In a footnote Thomson illustrated his perplexity. A bar of metal, he noted, will conduct heat from a hot body to a cooler one without producing any mechanical work, whereas passage of the same amount of heat through a Carnot engine will produce work. Carnot himself had at least indirectly made the same point, but either was not troubled by it or left it as a matter for later consideration. Thomson, however, saw a problem: “When ‘thermal agency’ is thus spent in conducting heat through a solid,” he asked, “what becomes of the mechanical effect which it might produce? Nothing can be lost in the operations of nature—no energy can be destroyed. What effect then is produced in place of the mechanical effect which is lost?”

Here, remarkably, Thomson was tiptoeing around a universal law of conservation of energy (using a word, indeed, which had hardly any currency and no precise meaning at the time), yet he didn’t seem to fully grasp the significance of what he was saying. Joule, a couple of years earlier, had argued that all forms of energy can transform, in suitable circumstances, one into another, but that energy as a whole cannot be created or destroyed. In his footnote Thomson seemed to concur—except that in the body of the paper he held fast to the rule that heat could not be transformed into mechanical work. Thus he was left clinging to two contradictory propositions.

Another inconsistency showed up. Thomson tried to harmonize the thermodynamic implications of Carnot’s theory with new measurements

by Regnault on the heat absorption capacity of steam at different temperatures and with Joule’s evidence for the conversion of work into heat at a constant rate. As before, he assumed that the efficiency of a Carnot cycle was the same at all temperatures. He found it impossible to establish consistency. In particular, his calculations told him that work should turn into heat with a conversion factor that was not constant but varied with temperature.

To a reader with some knowledge of physics and the benefit of hindsight, the most inexplicable flaw of Thomson’s 1849 paper is that he had already seen the answer to this last puzzle but had failed to absorb it. A few weeks earlier Thomson had written to Joule describing some of his calculations and expressing consternation that the results didn’t seem to match up. In a letter dated December 9, 1848, Joule, no mathematician, had casually thrown out the resolution. Casting his eye down Thomson’s lists of numbers, he observed that if one assumed the efficiency of the Carnot cycle to be inversely proportional to temperature, rather than constant, then everything fell into place. Work would convert into heat at a constant rate, as Joule had long argued. But Thomson would not let go of his interpretation of Carnot and therefore did not give Joule’s proposal the consideration it merited.

To avoid more confusion than we have unwisely waded into already, it is helpful at this point to jump forward a year, to 1850, when the German physicist Rudolf Clausius published his answer to these difficulties. Clausius, incidentally, noted that he not yet seen a copy of Carnot’s original paper and was relying on the expositions by Clapeyron and Thomson. His solution seems ludicrously simple, not to say obvious. In a Carnot cycle, he argued, some of the heat passes from hot to cold unchanged, but some is converted into work. Moreover, the relative proportions are such as to reproduce Joule’s suggestion that the efficiency of the cycle is inversely proportional to temperature.

This was staring Thomson in the face when he read Joule’s letter of December 1848. So impressed was Thomson by Carnot’s conclusions, it would seem, that he feared scrutinizing the assumptions too closely in case the whole elegant piece of reasoning should fall apart. Clausius understood perfectly well the problem, but took a very different attitude. “I believe we should not be daunted by these difficulties,” he wrote in his

1850 paper. “Then too I do not think the difficulties are so serious as Thomson does, since even though we must make some changes in the usual form of presentation, yet I can find no contradiction with any proved facts. It is not at all necessary to discard Carnot’s theory entirely.”

As Clausius went on to explain, Carnot’s general conclusions still held true even when his assumption about the nonconvertibility of heat was amended. Thomson was fully capable of seeing this. Apparently he never looked.

This failure, moreover, stands in contrast to the flexibility Thomson had shown in other cases, for example in his reconciliation of Faraday’s portrayal of electricity and magnetism with the apparently very different picture of action at a distance. In that case he sifted what was important and necessary from what was extraneous and incidental, and reconciled the two views. In the case of Carnot and Joule, he could not see beyond the apparent contradiction to the underlying consistency. It emerged some years later that Carnot himself, before his premature death, probably from cholera, in 1832, had seen that his assumption was wrong. In notes discovered only much later, he had written that “wherever motive power is destroyed, there is a simultaneous production of an amount of heat exactly proportional to the motive power that is destroyed. Conversely, wherever there is destruction of heat, motive power is produced.” This is precisely Joule’s position, reached a decade before Joule began his justly celebrated experiments.

As far as the science of thermodynamics is concerned, these questions are of no great consequence. Laws were established, it hardly matters by whom. Thomson is one of several people associated with the birth of this new science. He could easily have been, after Carnot, the most influential. Hindsight is dangerous, of course. Especially in science, everything is obvious once someone has figured it out. Still, Thomson’s stubbornness in sticking with Carnot’s false principle and doubting Joule—perhaps for no greater reason than that Carnot’s theory became lodged in his brain first—is an indication of a certain lack of flexibility, or an inability to take a leap in the dark, that inhibited Thomson’s scientific imagination.

***

Clausius was not alone in suggesting how to amend Carnot’s argument. William John MacQuorn Rankine was a Glasgow engineer who had, like Thomson’s brother James, learned his science through a mixture of schooling and practical work. In 1850 and 1851 he wrote a couple of long, tortuous, occasionally confused yet remarkably inventive papers that approached thermodynamics from a different perspective. Rankine had long been impressed with the idea of heat as a form of motion, and he took up an essentially atomic or molecular view. He conceived of a gas as a collection of atoms and supposed that heat was nothing but the motion of these atoms. This would be precisely the modern view were it not that Rankine settled on a model of atoms as tiny vortices, so that heat was essentially the rotational energy of these little whirlpools. Nevertheless, having made heat explicitly a kind of atomic motion, Rankine found it obvious that heat could turn into mechanical work, since both were merely different kinds of motion.

Rankine reached the same conclusions that Clausius did about the proportion of heat converting into work in a Carnot cycle. But because his reasoning sprang from a highly debatable atomic model, the generality of his conclusions was far from evident. Clausius, in fact, suffered from a lesser version of the same problem: He had implicitly assumed an ideal gas as the working substance of the engine he analyzed, and in places he relied, or appeared to rely, on his own model of a gas as a collection of particles in motion.

With the essentials of thermodynamics now in place, though marred by dubious assumptions and gaps of reasoning, Thomson’s particular intellectual powers came into their own. As he had done with electricity and magnetism, Thomson wrote over the next couple of years a series of long papers, “On the Dynamical Theory of Heat.” Like much of Thomson’s work, these papers represent an inextricable mix of originality and exposition. To some extent he simply took the principles established by Carnot, Joule, and Clausius, and set them down in a systematic way. On the other hand, there are many places where he showed a profound sense of logical and mathematical rigor and employed it to derive thermodynamic relations that depend as little as possible on unwarranted assumptions about the nature of heat or the constitution of gases. This was Thomson’s standard way of constructing a theory, going back once

again to his reading of Fourier: Apply sound reasoning to empirical knowledge and thereby create a theory that was sweeping and general but at the same time founded on fact.

Thomson showed how thermodynamics rested on just two basic principles. The first, which he credited to Joule, was that in any interconversion of heat and work, complete or partial, the sum of both quantities remained the same. This is conservation of energy, also known as the first law of thermodynamics. His second principle, credited to Carnot with the essential modification by Clausius, was that an ideal engine, capable of working forward or backward, extracted the maximum possible amount of work from a given amount of heat.

It was still not altogether clear, however, how best to formulate this second principle. Carnot had originally argued from the fact that one could not create work out of nothing, which makes the principle of maximum efficiency appear to be a consequence of the first principle, that energy cannot be created or destroyed. But this is a spurious association, coming about in essence because Carnot stuck with the idea that heat and work were distinct. With the recognition that an engine converted heat into work, discussion of the efficiency of that conversion became a separate issue.

Thomson stated the second principle thus: “It is impossible, by means of inanimate material agency, to derive mechanical effect from any portion of matter by cooling it below the temperature of the coldest of the surrounding objects.” That is, a machine can derive work when temperature flows from hot to cold; it can’t produce work by making some object colder than everything else. This was not very different from what Clausius had said, though Thomson seemed to think his statement a little more profound. Mainly, it appears, he wanted to distinguish his own words from those of Clausius, to whom he gave generous credit—“the merit of first establishing the proposition upon correct principles is entirely due to Clausius”—which he instantly took back: “I may be allowed to add, that I have given the demonstration exactly as it occurred to me before I knew that Clausius had either enunciated or demonstrated the proposition.”

In his series of papers Thomson also picked up the question he had posed in a footnote the year before about the passage of heat by conduc-

tion from a hot body to a cold one without any concomitant production of work: This heat, he said, was “irrevocably lost to man, and therefore ‘wasted,’ although not annihilated.” The meaning of “waste” here was left hanging, but Thomson came back to it in 1852 in a short note with the striking title “On a Universal Tendency in Nature to the Dissipation of Mechanical Energy.” He wrote this after he had come across a famous essay, “On the Conservation of Force,”⁷ published five years earlier by Hermann von Helmholtz. Although this essay is sometimes credited with establishing the law of conservation of energy, Helmholtz mainly rounded up various arguments and bits of evidence from other authors to show that a universal principle indeed exists. He discussed energy of all forms—energy of motion, gravitational attraction, heat, even chemical—and argued that transformation among all these forms is possible, but never creation or destruction.

Thomson’s reading of Helmholtz crystallized a qualitative notion into an absolute rule. In his note “On a Universal Tendency,” Thomson built on his new understanding to argue that when heat is “wasted” but not lost, it distributes itself in such a way that no further work can be obtained from it. The ultimate fate of any system is for the temperature to become the same everywhere. Thomson introduced here a remarkable number of new ideas and definitions. First he made a distinction between “statical” energy, such as is possessed by a weight suspended at some height, and “dynamical” energy, which is the energy of motion when the weight falls. Rankine later introduced the terms “potential” and “actual,” and Thomson then substituted “kinetic” for actual. These are the modern terms.

He also sharpened the distinction between “reversible” and “irreversible” processes. Carnot had recognized that a reversible engine gives maximum efficiency, but he had apparently not stopped to wonder about the fate of the heat lost, through conduction or escaping steam, in a less efficient and therefore irreversible engine. In an irreversible process,

Thomson now explained, heat that moved from high to low temperature without creating work did not signify any overall loss of heat energy, but the possibility of obtaining work from that heat was gone forever.

Because reversibility was the ideal, never realized in practice, Thomson argued that all natural processes, probably including biological and animate ones as well as purely physical and chemical changes, represented a loss of potential work from heat. From this he jumped to a “cosmic” conclusion: “Within a finite period of time past the earth must have been, and within a finite period of time to come the earth must again be, unfit for the habitation of man as at present constituted, unless operations have been, or are to be performed, which are impossible under the laws to which the known operations going on at present in the material world are subject.”

The idea of the universe running down to a state of enervated uniformity has become known as the “heat death,” a name and idea often attributed to Helmholtz or sometimes Clausius. But Thomson clearly made the point in 1852, although he didn’t come up with the catchy phrase.

Finally, and a little sneakily, Thomson managed in the course of his several papers on heat and the new thermodynamics to work in a revision of his absolute temperature scale. He stuck to the essential idea of defining a temperature difference according to the amount of work produced by a Carnot engine, but having finally understood that the efficiency of a Carnot engine itself varied with temperature, he had to adjust his original argument. It was a straightforward thing to do, and his old temperature emerged as simply the logarithm of his new temperature.

In a note written in 1881 for the publication of the first volume of his collected papers, he made light of this adjustment, saying only that the new scale was “practically more convenient” than the old one. But there were two significant differences. First, the revised temperature corresponded to the temperature defined by an ideal gas thermometer and was therefore closely related to the practical temperature scales that laboratory scientists had long used. Second, where the old scale descended to minus infinity, the new scale had a zero. With his revised understanding of Carnot, Thomson could see that the efficiency of an ideal engine would approach 100 percent as the temperature dropped to absolute zero. Tem-

peratures below that would make no sense for a variety of reasons, among them that it would then seem possible to get more work out of an engine that was put in as heat.

It is thus in his 1852 paper that Thomson truly established the existence of an absolute zero of temperature. In the modern picture of heat as the motion of atoms, it is obvious that when all motion has ceased, heat is absent, and temperature can go no lower. But Thomson could not and would not avail himself of any such unwarranted assumption. It is a tribute to the power and thoroughness of his reasoning that he could deduce the existence of an absolute zero without any reference to the physical nature of heat itself. He used only what was known empirically of heat’s properties. Fourier would have been proud.

By this time the first law of thermodynamics was understood by all to be the conservation of energy. No one person can truly be credited with its discovery or enunciation. Many people made qualitative proposals for such a principle, on more or less philosophical grounds, whereas James Joule came to the idea through careful measurement and experimental test. Thomson, adamant that theoretical proposals without experimental support were next to worthless, became a great advocate of Joule, to an extent that generated controversy and unseemly attacks that swirled about for decades to come.

The origin of the second law of thermodynamics is murkier still. In modern textbooks the second law is a statement about a quantity called entropy. In reversible changes, entropy stays the same; in irreversible ones it increases. Entropy can never decrease. Therefore, the entropy of the universe as a whole is always rising. This is a more precise statement of Thomson’s cosmic conclusion of 1852.

One of the few undeniable truths about the second law is that the word “entropy” (from a Greek construction meaning “transformation”) was proposed by Clausius, in 1865, when he bestowed the name on a quantity he had first formulated in 1854. Rankine in 1850, however, had defined a “thermodynamic function” that looks very much like entropy, except that it was tied to his vortex model of atomic motions, and Thomson in his 1852 account of “Universal Dissipation” had written down an expression that, with a bit of adjustment and translation, can readily be identified with entropy. At the time, though, he evidently didn’t

feel the urge to isolate this new concept and give it a name. In any case, Rankine, Thomson, and Clausius, in these earlier papers, observed only that in reversible processes a certain quantity stayed the same. They didn’t yet delve into the significance of this quantity, if any, for irreversible processes.

If the second law of thermodynamics is simply the principle that reversibility implies maximum efficiency, then credit goes to Carnot. But Carnot was thinking only of engines, not universal processes; he was working with a false assumption about caloric that precludes definition of either a first or a second law of thermodynamics; and his own words suggest that he did not see a separate principle here, only an instance of the general prohibition of perpetual motion. Subsequently Rankine, Clausius, and Thomson all made contributions to the statement of a second law, both in its physical conception and in the mathematical demonstration of its universality. Clausius in the end christened the child and most often gets the credit.⁸

Scholars continue to debate, rather fruitlessly, who said what and when, and what their fumbling words meant. One lesson is that science is harder than it looks. Of Thomson’s participation, though, it is difficult to avoid the judgment that he didn’t do as much as he might have done. He had an exceptional ability to sort and clarify, to resolve confusion and contradiction, and many of the standard elements of classical thermodynamics trace back to his definitions and arguments. On the other hand, at crucial points he needed a prompt from someone else. He developed the full theory of Carnot engines only after Clausius had supplied the essential idea that heat was consumed, not just transferred from one place to another. He made use of a quantity that eventually became entropy but did so apparently without seeing the general utility of it, as if he found it convenient for a specific purpose but failed to look beyond.

In the 1850s thermodynamics was imperfectly understood even by its creators. Nevertheless it was abundantly clear that scientific under-

standing of heat and work and energy and their interrelationship was no longer a cause of qualitative mystery, but could be captured in a handful of precise mathematical expressions. William Thomson, both as originator and expositor, was unquestionably one of a handful of people who had turned ill-defined notions into a new and fundamental discipline of physical science. His work in electricity and magnetism, though not developed to the same degree, nevertheless gave to those subjects a range and coherence they had not previously possessed.

***

In the summer, Thomson often traveled to Bad Kreuznach in the Rhine valley, where he could hike and think, and his ailing wife could take the waters. She was not at all well. One year Thomson told his brother James that “she suffers much after the driving and walking and is quite unable to sit up without much pain in her own room…. Dr Johnson … says it will do good notwithstanding the pain & fatigue, to a limited degree, but he says she is not in a fit state for almost any exercise.” They went to Kreuznach a number of times, and Thomson took a little comfort in the fact that his wife was even allowed to try the iron waters there.

During his 1855 visit to Germany, Thomson arranged to meet Hermann von Helmholtz, whom he had admired since reading his influential 1847 essay on the conservation of energy. Like Thomson, Helmholtz had wide-ranging knowledge across all of science (he had begun in medicine, moved into physiology, learned physics and mathematics in order to understand the science of perception, then began a career in physics proper) as well as an ability to synthesize arguments and evidence into a coherent whole. Helmholtz had come to Britain in 1853 and made a trip to Scotland, before attending the British Association meeting in Hull, in order to search out Thomson. He recounted his fruitless journey to his wife: “From Edinburgh I traveled for a couple of hours in the afternoon through a heavily built-up hilly area, with a variety of ruins, to Glasgow. This is a very big (pop. 300000) industrial city, horribly noisy and busy, swarming with poor, red-haired, dirty, unhealthylooking workers. It did not make a pleasant impression. I was looking for a physicist, Prof. Thomson, who has worked a great deal in matters con-

cerning the conservation of energy, but he had gone away to the seaside, so I strolled about in the streets until I’d had enough, then came back.”

Two years later they succeeded in meeting. Helmholtz again recorded his impressions in a letter to his wife: “As he is one of the leading mathematical physicists in Europe, I expected to find a man somewhat older than myself, and was not a little astonished when a very youthful, exceedingly blonde young man, almost girlish, appeared before me…. He exceeds, I might add, all the scientific greats I know personally, in sharpness, clarity, and quickness of mind, so that at times I felt dull-witted beside him.”

Helmholtz’s surprise is understandable. Thomson had been publishing important papers for almost 15 years. He had established theorems in applied mathematics, extended Fourier’s studies of the flow of heat, clarified the relation of electric charges and magnets to the forces they produced, as well as the interaction of magnetism with currents, and done as much as anyone to establish the foundations of classical thermodynamics. No other scientist in Europe at the time could lay claim to such range and depth of achievement. He had celebrated his 31st birthday just a few weeks earlier.