Seventeen years old, William Thomson arrived in Cambridge at the end of October 1841 to begin his undergraduate studies. For a few days he felt adrift and aimless. With his father James Thomson, professor of mathematics at Glasgow University, he had made the tedious trip from Scotland to eastern England: mail coach from Glasgow to Carlisle, in northwestern England, where they spent the night; another coach across the country to the eastern seaport of Hull; across the Humber estuary in a little steamboat; and finally on to Cambridge in another coach, arriving in the late evening. Professor Thomson managed to get an inside seat, but William had to perch on top until they got to Bedford, 30 miles west of Cambridge. His father stayed a couple of nights and, an academic man himself, dined at the fellows’ table of St. Peter’s College while his son ate with the new and returning students. Then Professor Thomson returned to Scotland to prepare for his own classes, leaving his son to start his new life. William had roamed a good deal in Scotland and had traveled to continental Europe twice, but these had been family adventures, in the lively company of his three brothers and his older sisters, Elizabeth and Anna, with a maid and often a tutor, all under the strict guidance of their father. William was an affable, gregarious young man but unused to isolation.

“Since he has left I have had very little to do, since lectures have not begun yet, & I have not got any advice as to what I should read,” he wrote to Anna after his father’s departure. So he idled about, getting to know the town and St. Peter’s College (as it was usually called then; now it is known as Peterhouse). The grandeur of the place surprised him. “I had no idea there were such fine gardens and grounds about the Colleges,” he wrote to Elizabeth. Wryly he informed Anna that “to make the time pass less heavily, I have been going out every now and then, & coming in again.” He went on short walks beyond the city. Accustomed to hiking around the lonely lochs and spectacular mountains of Scotland, he found the flat, empty landscape around Cambridge disheartening, but soon he was out walking and talking with his fellow undergraduates. Conversation and burgeoning friendships pushed the dreary landscape into the background. With guidance from his sisters he learned to prepare a small breakfast in his college rooms. He asked Anna to tell him how much tea to put in his cup for breakfast. From Elizabeth he wanted to know whether he should put coffee in the water before it has boiled or after.

William’s outgoing nature soon asserted itself. He began to make friends with other new students. The college tutor, Dr. Cookson, called on the young men to attend a wine party in his rooms. “I made my appearance in fear and trembling,” he told Anna with comic exaggeration. He had grown up in a house where distinguished academics came and went, and knew how to behave with charm and civility. At Cookson’s party he fell into awkward conversation about “college, buildings, Fitzwilliam museum” and other small matters, all of which, he allowed, “was on the whole not unedifying.” His initial nervousness overcome, he was not going to be overawed by the chitchat of the dons.

After a few days like this, lectures started, and William was instantly busy. “I have got no time to be dull,” he now told Elizabeth, “as I have got as much to do as I can possibly accomplish, and more besides.” More rewarding than walking, he soon discovered, were the pleasures of rowing on the Cam. William knew all about boating on the clean open waters of Scotland. The narrow river in Cambridge—“an exceedingly muddy and sluggish stream,” he told Elizabeth—had traditions all its own. “I adventured myself to-day for the second time in a funny (or funey, or funney),

i.e. a boat for one or two people to row in. It is certainly rather a venture to go in them as we can hardly stand upright in them for fear of upsetting them.” Quickly he mastered the little craft. Though tempted, he shunned traditional Cambridge rowing, in eights, in which his college was doing well. “Rowing for the races is too hard work for getting on well with reading; and besides, the men connected with the club are generally rather an idle set.” Not only that, a few days earlier a foolhardy student from Queen’s College had managed to drown himself trying to shoot the modest three-foot fall on the Cam.

After a week or two, William’s unease dissipated entirely and he threw himself into college life. He went to his lectures, read voraciously, studied in his rooms, rowed on the Cam, tramped around the muddy fields with his new friends, and, like any smart undergraduate away from home for the first time, stayed up late talking earnestly of this and that. His already advanced knowledge of mathematics set him apart from most of his fellow students. And a secret about him soon came to light that made him the object of unconcealed awe.

Six months earlier, in May, the Cambridge Mathematical Journal had published a short paper correcting errors in a recent book by the Edinburgh professor of mathematics, Philip Kelland. Only the cryptic initials P.Q.R. identified the author of this concise and confident work. The editor of the Journal was David F. Gregory, a fellow of Trinity College. He knew P.Q.R.’s identity and was eager to meet the young man. And so a few days after his father left, William Thomson called on Gregory to discuss not only a brief addendum to his first P.Q.R. paper, which subsequently appeared in the November Journal, but also more advanced work that he had developed over the summer and wished to submit to Gregory’s editorial scrutiny.

Back in Glasgow, meanwhile, James Thomson had a visit from Archibald Smith, son of a local sugar merchant. Smith was about to return to Cambridge where he too was a Trinity mathematician, having graduated a few years earlier. He dropped in on Thomson after being startled to learn from Gregory that the author of the P.Q.R. paper was none other than Thomson’s son. He “asked your age, and was surprised you could have written it,” James Thomson wrote proudly to William. Toward the end of November, a bare month after William’s arrival at

Cambridge, Gregory and Smith came by his undergraduate rooms to talk about mathematics. “It was certainly a great honour for a freshman of St. Peter’s to have two fellows calling on him,” he wrote blithely to his father. “They staid, I suppose, nearly three quarters of an hour.” But success had come so easily to William, even at this young age, that he scarcely wondered at it.

***

The previous summer, 1840, James Thomson had taken all six of his children on a holiday to Germany. The older children combined recreation with education. They studied German with a tutor and went on hikes, good for body and mind. Their father pointed out geological features and notable plants along the way. Continental expeditions were unusual for the times and, for a large family, expensive and cumbersome. James Thomson was a thrifty man and only moderately affluent, but for the improvement of his children he would spend what he must. The summer before he had taken his clan to Paris to see the sights and to improve their French. Leaving the four boys behind with a maid and tutor, he took off with his daughters to explore southern Germany and Switzerland on horseback. Elizabeth was then 20 years old, Anna 19. Young women, he told his daughters, had little opportunity to travel by themselves, and he wanted to give them this chance for adventure while he could. Back home, sober Glaswegians regarded them as brave and unusual girls and invited them out to dinner to tell of the exotic sights they had seen. The trip was doubly memorable for Anna. At a hotel in the Grindelwald in Switzerland they ran into William Bottomley, who had been a student of her father’s when he had taught mathematics in Belfast, before moving to Glasgow. Anna stayed up late talking to this “delightful young man.” They married in 1844.

On these trips William Thomson halfheartedly kept a diary. Before going on to Paris, the family had stayed almost a month in London, where Robert, the youngest son, underwent surgery to remove a stone. This was 19th-century surgery, before anesthetics. The boy, 10 years old, was trussed up firmly on the operating table while the doctor opened him up. Elizabeth waited at the door, alarmed by his cries and moans, and rushed in to comfort him as soon as the ordeal was over. William and his father, meanwhile, spent the morning touring the British Museum.

The danger of infection, often fatal, hovered over Robert for several days, but he was quickly on the mend. In the meantime the other children saw the sights. William and Elizabeth were up and about early, feeding the ducks in St. James’s Park or exploring nearby Westminster Abbey. William recorded in his notebook a few of his impressions of London, revealing a pleasantly impish humor that largely disappeared from his later notes and correspondence. Once when he and his sister were in Hyde Park they heard that Queen Victoria was in the vicinity “and we accordingly determined to wait and see her. After waiting for about half an hour, we succeeding in catching a glimpse of the top of her bonnet, a most overwhelmingly interesting spectacle, so much so indeed, that I forget its colour.” A few days later, on his own, he spotted the royal carriage again, chased after it, and was amply rewarded: “As the Queen’s head was averted I had the inexpressible felicity of seeing her most gracious bonnet.”¹

On the way to Germany the following summer, William again made a few desultory notes. On May 21 the family went by steamship from Glasgow to Liverpool then to London by train and on to Rotterdam by steamer, and finally by steamboat down the Rhine to Bonn, arriving close to midnight on the last day of June. Then William put his notebook down and left the remaining pages blank.

But he had no need to write down his impressions of Germany. The few weeks he spent there remained bright in his memory until he died. There his career in physics began; there he acquired a perspective (it would be too grand to say philosophy) on mathematical science that he maintained, for good and ill, throughout his life.

***

In 1822 the Frenchman Jean-Baptiste Joseph Fourier published his great book Théorie Analytique de la Chaleur (Analytical Theory of Heat), one of the milestones of physical science. At that time, and for a couple of decades after, the nature of heat resisted understanding. Most natural

philosophers held that it must be some kind of tenuous subtle fluid. Heat flowed, after all, and could be stored up and transferred with some degree of control from one place to another. It resembled a fluid more than it resembled anything else. This putative heat-fluid acquired the name “caloric.”

But if scientists didn’t understand what heat was, they knew a good deal, practically speaking, about what it did. Fourier’s particular genius was to construct a quantitative mathematical theory of the behavior of heat on a foundation of knowledge derived from observation and experiment, so that ignorance of the nature of heat was no impediment. This was an innovation, philosophically as well as scientifically. Mathematicians had always worked in an axiomatic way—start with principles and deduce consequences—and when they came to apply themselves to natural phenomena, they aimed to apply the same style of reasoning. But if no principles or axioms were known, then how to begin?

Direct observation, on the other hand, showed that heat flowed more readily through some substances than others, was retained by some materials longer than by others, and flowed faster where the difference in temperature was greater. Building on these elementary facts Fourier saw how to create a general theory of heat flow. Imagining a three-dimensional body with heat distributed within it, so that it was hotter in some places, cooler in others, Fourier came up with the crucial concept of the temperature gradient. Just as water flows downhill along the steepest incline, so heat moves in the direction along which temperature falls most sharply. In turn, the movement of heat changes the temperature pattern within the body. Ultimately heat will flow within a closed body so as to iron out all differences. When the temperature is the same everywhere, there is no gradient at any point, and heat will no longer move about.

When he was just 15, William Thomson heard the Glasgow astronomy professor John Pringle Nichol praise Fourier during his lectures. He asked Nichol whether he should read the Théorie Analytique. “The mathematics is very difficult,” Nichol cautioned, but William got the book from the library anyway and set himself to understand it.

How easily he read and digested Fourier’s theory depends on which reminiscence we are to believe. In 1903, speaking at the unveiling of a stained glass window memorial to Nichol, the 79-year-old Lord Kelvin recalled that “in the first half of the month of May 1840 I had, I will not

say read through the book, I had turned over all the pages of it.” But to his biographer S. P. Thompson three years later he offered a more robust account: “On the 1st of May … I took Fourier out of the University Library; and in a fortnight I had mastered it—gone right through it.”

Either way, Fourier undoubtedly had a formative influence on William Thomson’s scientific thinking. He saw that the great virtue of the Frenchman’s work was that, despite his title, the book did not in fact provide a theory or explanation of heat. As Fourier proclaimed: “Primary causes are unknown to us; but are subject to simple and constant laws, which may be discovered by observation, the study of them being the object of natural philosophy…. The object of our work is to set forth the mathematical laws which [heat] obeys…. I have deduced these laws from prolonged study and attentive comparison of the facts known up to this time….”

In other words, it is not necessary to understand the true nature of a physical phenomenon; instead, one observes and measures how it behaves and devises mathematical laws accordingly. That such an analysis yielded powerful and general results struck Thomson with the force of youthful revelation. Fourier’s words became a mantra to him, the bedrock of his view of physics. He acquired a lifelong detestation of speculation or metaphysics. Any scientific proposal must be grounded in a combination of established principles and empirical facts, and must yield mathematically rigorous results. Fourier’s magnum opus was some 400 pages long, but his new theory occupies only a small part of the book. The bulk of the treatise consists of numerous calculations of heat flow in all sorts of geometries (blocks, rods, spheres, and so on) to show the versatility and universal validity of his methods. It is not so implausible after all that William may have “mastered” the volume in a couple of weeks: the elementary ideas, once grasped, make evident good sense, and the mathematically acute young man would not have found Fourier’s repetitious calculations forbidding.

Not everyone embraced Fourier. Philip Kelland, the Cambridge-educated professor of mathematics at Edinburgh, published in 1837 his own Theory of Heat, in which he claimed to find contradictions and inconsistencies sufficient to invalidate many of Fourier’s findings. A couple of days before the Thomsons left Glasgow for their trip to Germany, a copy of Kelland’s treatise came into William’s hands. He was, he later recalled,

“shocked to be told that Fourier was mostly wrong. So I put Fourier into my box, and used in Frankfort [sic] to go down to the cellar surreptitiously every day to read a bit of Fourier.” Surreptitiously, because he was supposed to be learning German, but when his father discovered that William was avoiding one lesson in order to delve into something much deeper, he could hardly be displeased. And any displeasure evaporated when William jumped from his seat one day and declared abruptly, “Papa! Fourier is right, and Kelland is wrong!”

Kelland stumbled not over Fourier’s theory itself but over a novel method he used to solve the equations of heat flow. The trick involves the construction of what are now called Fourier series. The vibration of a violin string furnishes a classic example. The fundamental musical note derives from the oscillation of the string as a whole—its ends fixed, the center point moving up and down with a fixed period. But then there are the higher harmonics: the center point stationary and the two halves moving in opposite directions, one going up while the other is going down; then the string divided into thirds, quarters, and so on. At any moment, the shape of the violin string represents the sum of a series of simple waves with successively smaller wavelengths.

Any smoothly varying mathematical function defined over some finite length can be likewise represented as the sum of an infinite series of waves with suitably chosen amplitudes. In his book, Fourier used this method to solve many examples of heat flow. He might imagine, for instance, a cylindrical rod, initially at the same temperature throughout, with one end abruptly brought into contact with some body at a lower temperature. Heat flows out of the rod, and a gradient develops along it. To determine the mathematical form of this changing temperature gradient, Fourier found it easier to calculate the components of a suitable sum of waves: a Fourier series. The technique has become a standard tool in applied mathematics.

Kelland didn’t understand it. Fourier was partly to blame. He constructed series in a number of slightly different but essentially equivalent ways and would jump from one to another, depending on which was more convenient for a particular problem, without always making it clear what he was up to. But William Thomson had the wit to see that Fourier reached the right conclusions despite his occasional sloppiness and

showed that the Frenchman’s slips and omissions were not fatal. Kelland, by contrast, simply saw the problems and stopped dead. Where Kelland displayed a pedantic sense of logic, Thomson demonstrated real insight. He showed himself more acute than Kelland and more rigorous than Fourier. He provided simple proofs of some assumptions that Fourier had made but not verified. This was bravura from anyone, let alone a boy who had celebrated his 16th birthday only a few weeks earlier.

William quickly convinced his father that indeed Fourier was right and Kelland wrong. James Thomson, a good if not original mathematician, saw that this was a substantial result. His son had provided a clear and reasoned decision in a dispute between two eminent men. His sharp analysis warranted publication in a mathematical journal.

Back in Scotland, father and son worked up a paper explaining Kelland’s errors, and early in 1841 James Thomson sent it off to Gregory, hoping it could appear in the Cambridge Mathematical Journal. Established just four years earlier by the young mathematicians Gregory and Archibald Smith, the Journal aimed to provide a venue in English for the new kind of mathematical physics that the French especially were developing. It was at James Thomson’s insistence that William disguised himself as P.Q.R. The letters had no particular meaning, except that they are often used as a triplet of variables in three-dimensional mathematical problems. This anonymity was ostensibly for his son’s benefit, as James Thomson apparently thought it inappropriate that a boy should publish openly in a scholarly journal. He may also have wished to spare Kelland, a fellow professor, the embarrassment of having his errors pointed out by a child.

As a matter of propriety, Gregory decided Kelland should know the name of his accuser and see the paper before it went into print. So James Thomson wrote directly to Kelland and received at first a cool response: “As to the insertion of the paper in the journal I think Mr G did quite right in corresponding with you first for two reasons. 1. Because an author never gets any credit for rectifying blunders. 2. Because the plain wording of the remarks is not quite what should appear in a periodical lest it should awaken the wrath of parties concerned & the blame fall on the editor.”

Kelland added some technical criticisms, and James Thomson, al-

ways ready to act the diplomat in pursuit of larger aims, agreed to remove some of the phrases that had irked the other man. He further mollified him by declaring that William’s “sole object is to establish what is true, and to remove any false impressions with regard to Fourier.”

Privately, to James Thomson, Gregory agreed that “the flippant manner in which Mr Kelland speaks of Fourier would deserve pretty strong terms of reprobation.” To his credit, though, Kelland saw the error of his arguments and the correctness of William’s and quickly agreed to publication. “I am very much pleased with it and think if he works it up well into a paper it will be most interesting,” he wrote, after James Thomson had soothed him. “Send my regards to your son,” he added, for his “great service to science.” Although Kelland played no further role in William’s life, his sister Elizabeth says that the two became good friends later on.²

James Thomson’s tact made what could have been an awkward scientific debut into a rather smooth performance. The paper appeared in the Cambridge Mathematical Journal of May 1841. Though admittedly written with James Thomson’s help, it displays an assured, straightforward manner. After describing briefly the problems Kelland raised in his “excellent Treatise on Heat,” William immediately showed that Fourier’s answers are right, even though some of his arguments appear patchy. “I have examined the other series given by Fourier, on this subject, and they seem all to be correct, with the exception of misprints and mistakes in transcription, which, unfortunately, are very numerous,” he wrote. In one case he gave a detailed argument to show that Fourier must have done a calculation correctly, even though some of the intermediate steps printed in the Théorie Analytique are wrong. There is nothing apologetic or obsequious about the paper. William states his purpose, writes out his calculations, and presents his conclusions. It is an adult work.

***

During his three undergraduate years William published a dozen papers in the Cambridge Mathematical Journal. He read eagerly and stud-

ied hard, at first with his college tutor Cookson and then for two years with William Hopkins, a highly regarded private tutor. He was never idle. For exercise he went on the river and strode for miles about the dull Cambridgeshire landscape. His fellow undergraduates would visit his rooms, or he theirs, and they would talk mathematics, dabble in political questions or other news of the day, discuss religion, perhaps touch on Shakespeare or the classics or lighter reading. He took up the cornopean (also cornopiston, from the French cornet à piston), a kind of French horn, and in 1843 became a founding member, later president, of the Cambridge University Music Society. Sometimes he spent too many hours on the cornopean and regretted that he hadn’t read as much mathematics as he might have. Sometimes he read for so long he needed to walk or row to refresh his mind. But always he got his work done. A lifelong habit of incessant activity took root.

Despite William’s precocious ability and prodigious achievements, James Thomson suffered from a constant fear that Cambridge would seduce his son away from a rigorous intellectual path into a dissolute and purposeless life of wine parties, rowing, and the reading of light novels. Born to a poor farming family in what is now Northern Ireland, James Thomson had doggedly used his intellectual talents to build himself a sound and solid life, resisting along the way any distraction. William, by contrast, overflowed with almost casual brilliance. He devoted hours to his studies but had nervous energy to spare. Rowing, walking, and music were not distractions but essential recreations.

To his father, though, these extracurricular activities represented time and effort not applied in laying the foundation of a secure career. For some significant proportion of Cambridge students in those days, undergraduate life was devoted mainly to forming friendships and connections, learning how to comport oneself at afternoon tea parties, playing rugby or cricket, boating on the Cam, carousing and drinking in the evening—anything but studying in earnest. There were no entrance exams to the university. Anyone who had money and preferably a helpful family connection could enroll as a student.

Through most of the 19th century, the Cambridge student body fell into three roughly equal divisions. About a third were the sons (no women, of course) of the gentry, with private means, a family Oxbridge

tradition, and no urgent concern to find a career or profession. These students generally left the university with an ordinary rather than an honors degree, which still bestowed on them the right to regale friends and family with stories of their time at Cambridge. Another third were poor students, clever but from meager backgrounds, who survived on scholarships or charity and struggled to live while they devoted themselves to studying. For such people Cambridge was a lifeline from poverty, to be grasped securely and never let go.

Constituting the remaining third were the children of what we would now call middle-class professionals: doctors, lawyers, clergymen, headmasters, and the like. William was of this group. James Thomson never ceased to worry that his son might through inattention or complacency slip back down the social ladder that he had so determinedly ascended. Brilliance was all very well, but what brought success in the world was the correct attitude. He bombarded William with cautions and admonitions:

In Cambridge he saw his son beset all around with temptations and perils that could upend his promising career and destroy his future at a stroke. Even ice skating was on the list of dubious recreations. In December 1841, learning that William had been out on the frozen river, his father let fly with a paragraph that jumped from one lively fear to the next. He was concerned for William’s safety on the ice, he wrote, but hoped “farther, that it will not lead you into company that will injure or relax your moral feeling. I am sorry to hear that you have been boating—not on account of the thing itself, as I think there can be no danger, but

that you may be brought into loose society, a thing that would ruin you forever. I find that Ayrton [another Scottish undergraduate] goes to no wine parties, because of the excesses and other evils to which they lead. At present, I do not say, you should go to none, unless with fellows; but you should scarcely go to any others; and if you do go, observe the strictest caution, and always tell me about any thing you find. In more advanced years, you will see that my cautions are well founded.”

William replied with casual reassurance: “With regard to wine parties, I have gone to as few as I possibly could, and at any to which I have gone there has not been the least approach to excess…. I have given no wine parties, or indeed any parties yet, but I suppose I must return some of the invitations next term.” Although he didn’t row with the college team—“an idle and extravagant set,” he agreed—he took to the water as often as he could, reassuring his father that “I always row by myself in a funny, (or as it is called skulling…) or at least go in a two-oared boat, with some friend with whom I should otherwise be walking.”

William had a fine sense of what he could get away with. In February 1842 he wrote to his father with the startling news that he had, without permission, spent all of £7 on a secondhand boat, “built of oak, and as good as new.” This extravagance, he claimed, actually represented fiscal astuteness, since he would no longer need to hire a boat. His father was shocked by this insubordination. “You are quite right in anticipating that I would be surprised,” he wrote. “You allowed yourself to be cajoled and probably cheated…. Seven pounds for a tub that will hold only one person!!!” He called his son “a soft freshman” for being duped.

He threatened to make William return the boat immediately and get his money back. But with his remarkable and gifted son, the instinct to be firm ran up against a habit of indulgence. He consulted Cookson, the tutor, on the pros and cons of boating, evidently received a favorable opinion, and grudgingly paid up. The money came to William in a letter from his younger brother John, who reported the reaction of his mischievous sister Anna.

“I hope [the boat] is to your liking but it is not at all to Anna’s as she would like exceedingly that it were broken up, for firewood, or employed for a washing tub, as, till that time she will be constantly on the look out in the obituaries for the drowning in the Cam of an extraordinarily clever,

young Cantabridgian: and, if I say to her that you could surely swim across the Cam, she says that I know quite well that you might take the cramp.”

A month later James Thomson had softened entirely after hearing again from Cookson, who wrote “so favourably and so kindly regarding you” that he sent another £10 for the boat and other expenses. He apologized a little for his “admonitory style” but told William that “at your period of life, and placed as you are among many persons of different characters and habits, you require to be most circumspect, and to be firm in your adherence to what is right and proper, and in resisting every advance to what is bad.” Rowing itself, a vigorous and manly activity, James Thomson could not honestly object to. It was the lurking fear of loose morals and roguish young men that animated his concerns.

James Thomson’s numerous, lengthy, and repetitious letters to his son make him seem cautious to a degree, humorless, and puritanical. But he was not in person as dour as all that. John Nichol, son of Professor Nichol and playmate to the younger Thomson children, recalled Professor Thomson thus: “Good-hearted, he was shrewdly alive to his interest, without being selfish, and would put himself to some trouble, and even expense, to assist his friends. He was a stern disciplinarian, and did not relax his discipline when he applied it to his children, and yet the aim of his life was their advancement…. He was uniformly kind to me, and I owe him nothing but gratitude.”

His lowly origins colored James Thomson’s personality and anxiety over his children’s future. Born in 1786 on his parents’ farm near the small town of Ballynahinch, County Down, he was the great-great-grandson of a John Thomson who had fled religious persecution in the lowlands of Scotland around the time of the English Civil War, at the end of which, in 1649, Oliver Cromwell had sent Charles I to the executioner’s block. In Scotland, Protestants at first sided with Cromwell against the overbearing Charles, then against him as he too attacked Scottish political and religious traditions. Many Scots fled to the northern counties of Ireland.

Agnes King, daughter of William’s sister Elizabeth, says that the Thomson family was of “the fine old stock of Scottish Covenanters.” These were adherents to the King’s Covenant, which James VI of Scot-

land (later James I of England and father of Charles I) had signed in 1580 in formal renunciation of the Pope and the Catholic Church. The religious ramifications in Scotland of the Civil War in England verge on the incomprehensible (as well as the Covenanters, there were the Protesters, the Remonstrancers, the Resolutionists, and others), which is one reason the repercussions linger in Northern Ireland to this day.

Some of the Thomsons moved on to America. Others, James’s ancestors, stayed where they were even after Scottish affairs had quieted but continued to think of themselves as ancestrally Scottish. Over the years their religious ferocity abated into mainstream Presbyterianism. In 1798, when James Thomson was a boy, he witnessed bloodshed at the Battle of Ballynahinch, when English soldiers put down a brief Irish rebellion inspired by the recent French and American revolutions. His family provided food for the rebels, but the insurgents were quickly and easily defeated.

James Thomson acquired a fierce disgust of religious favoritism and sectarianism. He devoted his considerable talents and self-discipline to the furtherance of his own life, and though he was firm in his principles he always aimed to resolve difficulties by diplomacy rather than protest.

James received a little education from his father but went on to teach himself from books and later enrolled at a local Presbyterian school, taking in the standard improving diet of classics and mathematics. He was unusually bright, but even more remarkably diligent. While still taking higher classes at the country school, he served as assistant teacher to the lower forms. Later he became a master at the school in the summer months and for six years sailed to Glasgow every autumn to attend university there, which ran for a single long session from November to May. He obtained his M.A. in 1812 and two years later became a teacher of mathematics, geography, arithmetic, and bookkeeping at the Belfast Academical Institution. The following year he became professor of mathematics. In 1817 he married Margaret Gardner, daughter of a merchant family, whom he had met in Glasgow. He built a house in Belfast and started a family. Elizabeth came first, in 1818, then Anna in 1820, James in 1822, and William on June 26, 1824. Three younger children followed: John in 1826, Margaret in 1827, and Robert in 1829.

He began writing textbooks, not only on elementary arithmetic but

also on geography, astronomy, and more advanced mathematical subjects, including algebra, geometry, trigonometry, and differential and integral calculus. He had a way of bringing mathematics alive through illuminating examples, and his books became standards in many schools and colleges. They sold well enough to substantially increase his income, and some remained in print for decades. As late as 1880 his sons James and William together edited the 72nd edition of his Treatise on Arithmetic in Theory and Practice. In the preface (written for the 23rd edition of 1848), James Thomson senior decried the aridity of teaching by rote memorization, by which mathematics, “peculiarly fitted to call forth and improve the reasoning powers, is degraded into a dry exercise in memory.”

The liveliness and effectiveness of his teaching brought him renown. He lavished the same care and attention on the education of his children. He rose at four in the morning to work on his textbooks, taught at the institution during the day, and in the evening tended to his offspring. He began by reading to his children from the Bible and the classics, and introduced them, the girls as well as the boys, to arithmetic, geography, botany, and other elementary subjects. He resorted to tutors only for music, dancing, and French. As the children grew older, he would read to them from newspapers and magazines of current events and encourage them to comment on both style and substance. The older boys, James and William, he particularly encouraged in mathematics, and both proved quick—William quicker than James, James more thorough than William. At the ages of 8 and 6, the boys took a few classes at the Belfast Institution and took the top two prizes. In a presentiment of what was to become a common pattern, William came first ahead of his older brother. Elizabeth scrubbed and washed the wriggling William and dressed him in white trousers, black jacket, and tie, leaving James to dress himself similarly, then proudly marched them off to the institution to receive their awards.

James Thomson’s devotion to his children redoubled after his wife died in May 1830, having never regained her health following Robert’s birth. During their mother’s decline, Elizabeth recalled, James Thomson strained to keep his grief to himself and present only a sturdy figure to his children. Once, unseen, she saw him emerge from their mother’s bedroom and was “frightened to see my beautiful father, so tall and strong,

standing outside the door pressing his head against the wall.” The moment passed. He collected himself and went back to his wife’s bedside.

That very evening the five oldest children were summoned to their father’s study. “He was sitting there alone, at the side of the fire,” Elizabeth recalled years later. “As the little troop came into the room, he opened his arms wide, and we ran into them, and he clasped us all to his heart. I was the tallest, and his head dropped on my shoulder, and he said with a choking voice ‘You have no Mamma now.’ He held us a long time so; his whole breast heaving with convulsive sobs. Then he gathered the two little ones, William and John, on his knees, and kept his arms tight round us all,—his head resting on the cluster of young heads pressed closely together; and there we remained in silence and darkness, except for the glow of the dying embers, till at last the nurse came and asked leave to put us to bed.”

His youngest daughter, Margaret, had never been well and died the following year, not quite four years old. From this time on James Thomson was “both father and mother to us, and watched over us continually,” Elizabeth said, although she herself, as the oldest child, took on a maternal role and was always a more serious girl than playful Anna. To William, not yet six, the death of his mother only briefly darkened the happy progress of his childhood. He recalled nothing of her in later years.

The following year, 1832, the clan moved to Glasgow where James took up the professorial position he held until he died. He and his family got off to a difficult start. That year cholera raged through the city, as it periodically did, bursting out of the slums to threaten the whole population. The university at that time was near the old center of Glasgow, in an area that had been engulfed by cheap tenements for the inrushing factory workers while the more affluent Glaswegians drifted west. The mushrooming industrial cities of Great Britain all had their share of disease and squalor and drunkenness and crime, but Glasgow was among the worst. Many years later, when the university had moved to a new site, those who had known it in the old days could afford to let nostalgia color their recollections. As one long-serving professor recalled many years later:

The Thomson family took up residence in one of the 11 faculty houses forming a tight quadrangle known as College Court. It was a “dingy old place,” John Nichol remembered. They hunkered down for weeks until the cholera had burned itself out. Elizabeth recalled with a shudder the dead-cart taking bodies away at all hours. On top of this James Thomson discovered that his regular salary was far less than he had expected. Instead his income came largely from fees collected from the students who attended his classes. Few came at first to hear the new professor, and family legend records that his Glasgow position, far from solidifying Thomson’s entry into the professional classes, cost him money for the first year.

But for James and William, the arrival in Glasgow marked the beginning of their intellectual lives. Huddled in College Court, they made the acquaintance of other academic families, notably that of John Pringle Nichol, who introduced William to Fourier a few years later. John Nichol, the professor’s son, remembered Elizabeth and Anna as “both clever, good talkers and sketchers.” One of them (he diplomatically doesn’t say which) was “very pretty.” In sketches done by Elizabeth around this time both girls look charming, though the artist gives herself a slight edge. With the four boys they formed “a pleasant and happy group,” according to Nichol.

William and James began to sit in on their father’s classes. If their fellow students were surprised to see an 8-year-old in their ranks, they were astonished when the professor posed a difficult question that left the class silent except for the small fair boy who jumped up from his seat pleading, “Do, papa, let me answer!” He had always been a blessed child, so it seemed to Elizabeth. He was a bonny baby, fair-haired and blue-eyed. In Ireland a local artist had borrowed him one day as a model for an angel, suitably adorned in frills and ribbons. As a 2-year-old he was once

discovered sitting on the floor, staring at his reflection in a mirror and cooing to himself, “P’itty b’ue eyes Willie Thomson got!”

James Thomson doted on this most adorable of his sons, as Elizabeth records in a curious passage from her memoir: “William was a great pet with him—partly, perhaps, on account of his extreme beauty, partly on account of his wonderful quickness of apprehension, but most of all, I think, on account of his coaxing, fascinating ways, and the caresses he lavished on his ‘darling papa.’ When our father came in he would run to him, and jump about him like a little dog, exclaiming, ‘Oo’s nice good pretty papa, oo’s nice good pretty papa,’ and when his father stooped to greet him, the child would fling his arms about his neck and smother him with kisses, and stroke his cheeks endearingly. He had not words adequate to express his affection, and tried every conceivable way to make it felt. And this was not occasional demonstration; it was his constant habit, and had been from infancy. Sometimes the others thought there was a little affectation in this, especially when he used baby language after he could speak quite well; and we laughed at him, but he never heeded.”

This odd behavior, Elizabeth claims, excited no jealousy or resentment among the other siblings. William was a sweet-natured child. His siblings were proud of their beautiful and bright brother and pleased that he brought such obvious happiness to their recently bereaved father. In 1834, when William was 10 and James 12, they enrolled formally as Glasgow University students and frequently won the top prizes in their classes, in classics as well as mathematics. Most often, as in Belfast, William came first, James second. Nonetheless, William did not become spoiled or vain. His tutors and fellow undergraduates at Cambridge recalled him as a charming and sociable young man. “A most engaging boy, brimful of fun and mischief, a high intellectual forehead, with fair, curly hair and a beauty that was almost girlish,” recalled one contemporary years later.

Consciously or not, William learned through his childhood how to use his charm and his father’s affection to get his own way. After his unauthorized purchase of the “funny,” he frequently reminded his father how favorable rowing was for his health and therefore also his studies. “I have been reading moderately, and skulling a good deal in this vacation, so that every one tells me I am looking much better than I did some time

ago. Today, just before Hall, I returned from a skull of fourteen miles… and I am not in the least tired, but I shall be in excellent condition for reading in the evening,” he reported, and a few days later added, “I find that I can read with much greater vigour than I could when I had no exercise but walking, in the inexpressibly dull country round Cambridge.” About this time he recorded his weight as 8 stone 10 pounds (122 pounds) in his rowing jersey.

These little reminders paved the way for William telling his father, in May 1842, that he had used more of his tuition money to buy out his partner’s half-share in the boat. His fellow undergraduate, he explained, hadn’t rowed very much and when he did had damaged the boat and broken an oar, so that as before William’s new expenditure was in truth an economy. “I am sure you will perfectly approve of that way of spending the money since I have found the skulling, after two or three months trial, to be most beneficial to my health and reading,” he confidently asserted. His father grumbled, then paid up.

***

William’s reading focused, of course, on the study of mathematics, the exception being an irksome examination colloquially known as the “little-go,” which all honors students had to pass in their second year in order to demonstrate at least a passing acquaintance with Latin and Greek authors as well as works of a general religious or philosophical flavor.³ For this William boned up on a section of the Aenead and a little Xenophon, recording in his diary that he had been practicing on the cornopean a good deal “to relieve my head from the seediness concomitant upon littlego subjects.” Despite some fear that the classics would trip him, he easily negotiated the little-go. Then it was all mathematics. For the second and third years he studied mainly with his private tutor, or coach, William Hopkins. He had two goals. He wanted first to learn as

much advanced mathematics as he could, to build on what he had learned from his father and by his own initiative and develop a wide-ranging and systematic command of the subject. James Thomson had his doubts about Cambridge mathematics—there was an excessive reverence for Newton and a consequent resistance to the new ideas coming mainly from France—but even so, it represented the pinnacle of mathematical attainment in Great Britain. There was no doubt in his mind that a Cambridge mathematical education was what his son needed, but still, there was some anxiety about entrusting the youngster’s ripening brilliance to hands other than his own.

The second goal was to become “wrangler” for his year. This is the undergraduate who gets the highest marks in the mathematics honors examinations—the tripos, as Cambridge exams are still known, not from any tripartite nature of the subject examined but from the precarious three-legged stool on which examinees in olden days had to sit. Candidates for the mathematical tripos sat (in the 19th century at ordinary chairs and desks) a grueling series of eight lengthy papers undertaken over a period of six days. Nervous and sweating examinees were expected to spill out, in coherent manner, the most arcane and involved elements of the subject they had digested over the previous years. The top man was senior wrangler, the second junior wrangler; positions were reported down to 10th or even 20th wrangler. The London Times published the list. Being wrangler was a moderate sort of national honor as well as a university distinction.

Like all examinations but to an extraordinary degree, the competition for wrangler was a test of genuine knowledge, power of recall, concentration, nerves, and handwriting. Over the years it had acquired the qualities of a ritual, like the compulsory figures section of the Olympic ice-skating competition in which contestants must perform prescribed moves and jumps with precision and control. Technical mastery rather than originality or flair was the key. The well-coached wrangler candidate knew his essential mathematics but also knew how to write out stock answers to standard questions as concisely and rapidly as possible, in order to do as many problems as he could in the time available. It did not help, in the heat of the exam, to start thinking of more profound or comprehensive solutions than the one the examiners were looking for.

Nor was there any reward in perceiving valuable generalizations or wider implications of a narrowly defined answer. Compact, tidy handwriting was an asset. William Thomson wrote in a large though readable scrawl.

In the middle decades of the 19th century, because good teaching was rarely a high priority of the colleges, the system of mathematical coaching developed to a fine degree in response to the demands of wrangling. As a married man, Hopkins could not be a college fellow, but in the end he acquired a far greater reputation, not to mention a better income, as a coach than he would have done in a formal appointment. The sporting analogy is apt: A good coach secured high places in the exams for his pupils and thus attracted the better students in subsequent years. Hopkins charged £72 per student per year for twice-weekly sessions, might coach 10 or 12 students in total, often took additional classes during the long summer vacation, and thereby easily earned £800 a year or more—considerably more than the typical college fellows and a solid upper-middle-class income.

As with coaching for gentlemen who wished to row, the aim was to develop strength and stamina and the ability to perform reliably and repetitively strange motions that neither body nor mind would take up naturally. Coaches were generally men who had placed well in the wrangler competition in years past and who had a knack not only for training young men in the same art but also for predicting from one year to the next the questions that were likely to come up. Problems that would test the wranglers-to-be could come only from certain advanced branches of mathematics, yet they had to be solvable in the allotted time. An apt question, like a nifty crossword clue, was a praiseworthy construction in itself.⁴

Peter Guthrie Tait, another Scottish student who became a close friend of William Thomson, was senior wrangler in 1852. As one who had survived with distinction a difficult and painful ordeal, he was later scathing about the Cambridge system. “College Tutors and Lecturers take

but small part in the process of education,” Tait told the students and faculty of Edinburgh University in 1866, where he was a professor. “Private Tutors, ‘Coaches’ there, ‘Grinders’ we should call them, eagerly scanning examination papers of former years, and mysteriously finding out the peculiarities of the Moderators and Examiners under whose hands their pupils are doomed to pass, spend their lives in discovering which pages of a text-book a man ought to read, and which will not be likely to ‘pay’. The value of any portion as an intellectual exercise is never thought of; the all-important question is—Is it likely to be set? I speak with no horror of, or aversion to, such men; I was one of them myself, and thought it perfectly natural, as they all do. But I hope such a system may never be introduced here.”

The wrangler system, over the years, acquired a patina, a hushed mystique into which all new aspirants must be inculcated. J. A. Fleming, a Cambridge undergraduate in the late 1870s who became a pioneer of the pretransistor electronics industry, recorded in appropriately flat prose his experience of studying for the mathematical tripos. The student would visit his tutor, Fleming recounted, “at an appointed time, and the ‘coach’ gave him an examination paper of questions and supplied a hint or two as to how the questions were to be solved, and also marked certain chapters or parts in a text-book to be read…. Then the student went back to his own room and tackled the paper of questions, and read as requested. The next day we took our results to the coach, who noted successful answers, and gave a further hint as to the solution of unsolved problems. The coaches had great experience in forecasting the kind of question likely to be put in Tripos exams, and it would have been quite impossible to obtain a high place without their aid.”

One can easily imagine this exchange taking place in a sepulchral silence, sheets of carefully annotated paper gliding back and forth across a polished table to the accompaniment of restrained gestures and indications, as if it were part of a training program for novices in some secretive and highly regulated religious order. The greatest proportion of senior and junior wranglers, as it happens, went on to careers as ministers in the Church of England. Mathematics afforded few professional opportunities, beyond a few Oxbridge fellowships and a clutch of professorial chairs at the four Scottish universities—that or schoolmastering. In any case

advanced mathematical training was seen more as a kind of general strengthening of the mind than it was as preparation for a life of work with numbers and equations. The same was true of the classical tripos. Students of the classics learned by heart great stretches of Caesar and Cato and Ovid and Horace, and acquired the ability to ad lib a suitable Latin ode on any occasion. Men trained in this way were regarded as having intellects honed to less demanding tasks, such as running Her Majesty’s government and directing the course of empire.

William Hopkins was a superlative mathematical coach. Merely seventh wrangler in 1827, he had by 1849 coached 17 senior wranglers and 44 top three places; his pupil E. J. Routh, senior wrangler in 1854, became a coach with 27 senior wranglers to his credit. Those who gained the top handful of places each year could, if they wished, find a pleasant college fellowship or work as a coach and then fashion a career producing more wranglers. Thus did wrangling perpetuate itself over the generations, and as often as not it was the competent but less imaginative men who went on to become fellows and coaches, while those with some other ability besides that of writing out long mathematical answers at great speed took up other careers.

Hopkins was an exception to this dreary practice. He had genuine scientific ambitions. He pioneered the application of quantitative mathematical methods in geology, which had until then been largely a descriptive science, like botany. He analyzed the earth’s orbital motion, its rigidity and interior dynamics, the movement of glaciers, and most notably the distribution of heat within the earth. To his pupils he brought not just the tools for doing well at the tripos but also a deep appreciation of the nature of scientific problems and the use of mathematics in solving them. In the middle of the 19th century, the relevance of mathematics to science in general, as opposed to a few highly specific areas of physics, was by no means commonly accepted. Hopkins saw how rational analysis could be brought to bear on all manner of questions—but in the case of his exceptional pupil William Thomson, this was a superfluous lesson.

Even before he began his coaching with Hopkins, William had published a third paper in the Cambridge Mathematical Journal of February 1842. Where his first two papers, correcting Kelland’s misimpressions of Fourier, were works of mathematics, his third was recognizably a piece of physics, and a sophisticated piece at that. Written in Scotland in August

1841, before he arrived in Cambridge, it owed nothing to Cookson, Hopkins, or his other tutors or advisers. In it he described an analogy between the heat flow formulation of Fourier and the way an electric field spreads across space between charged objects. The Frenchman Charles Coulomb had established in 1785 that the electric force between two charges, like the gravitational force between two masses, decreased in proportion to the square of the distance between them. Subsequently the self-taught English genius Michael Faraday, son of an impoverished London blacksmith, had devised an alternative portrayal of electrical interactions, employing what he called “lines of force.” In Faraday’s vivid imagination (he knew no mathematics to speak of and substituted an acute, largely pictorial way of understanding physical phenomena), forces between electric charges were conveyed along curved lines, something like intangible elastic strings; these lines, moreover, repelled each other and so distributed themselves as economically as possible through space, creating a tension that we now recognize as the electric field.

These were vague notions, and William Thomson remained skeptical for some time of Faraday’s powerful but allusive insights. Nonetheless, he proved in his 1842 paper that, with physical quantities suitably redefined, he could use Fourier’s mathematics of heat flow to portray the geometry of Faraday’s lines of force. This was more than mathematical cleverness; it hinted that electric force “flowed” through space just as heat flowed through matter. It turned out, moreover, to be a powerful way of analyzing electric forces. Coulomb’s inverse square law was fine for dealing with the simple case of two discrete electric charges but became intractable if one wanted to investigate more complex geometries—the force between an electrically charged sphere and a flat plate, for example. Fourier’s treatment of temperature distributions became in William’s adaptation an equally general way of dealing with distributions of electric charge. The method (as subsequently developed by Thomson and others) is still taught today. For so young a man to have devised it when both heat and electricity were so poorly understood was a remarkable step.

A fourth paper for the Cambridge Mathematical Journal was the first that William produced from Cambridge. Again it displayed sharp physical insight. The pattern of temperature within some body, Fourier had shown, would always become more uniform as time passed. Conversely, William realized, temperature must become less uniform, more irregular,

as one turned the clock backward. He proved a striking result. A pattern of temperature in some object cannot in general be calculated backwards in time without limit, because mathematically impossible distributions of heat arise. To turn it around, a heat distribution existing at the present moment can be the outcome of an initial arrangement that existed only some finite time ago. This was a straightforward mathematical demonstration, but it was not long before important physical applications came to William’s mind. Most notably, he applied this reasoning to the present state of heat within the earth (so far as it was known) and reached the conclusion that the earth could not have an unlimited past. This seems uncontroversial today; in the middle of the 19th century it was not. Even the application of mathematical reasoning to “cosmological” questions such as the origin of our planet struck many Victorian minds as close to sacrilegious.

While he was cramming relentlessly for the wranglership, William kept up a remarkable rate of publication. The subjects he had broached—the flow of heat, the geometry of electric fields, and the mathematical parallels between the two—formed the seeds of work that he developed much more deeply in the first part of his scientific career. It all rested on Fourier’s principle of formulating mathematically sound arguments relating to observed phenomena. As his friend P. G. Tait remarked many years later, “Fourier made Thomson.”

***

One of William’s fellow undergraduates, reminiscing years later, reported that the startlingly accomplished young man was being touted as a senior wrangler just days after his arrival at Cambridge. William himself suffered doubts from time to time. For a few months in early 1843, halfway through his second year, he kept an intermittent diary of his undergraduate routine and habits, and more interestingly of his fears and anxieties.⁵ A student from Germany named Ludwig Fischer aroused concern: “I must read very hard and try to be at least as well prepared as he

is,” William noted on February 15. He felt no lack of intellectual fire-power but worried that his incomplete education before Cambridge left him at a disadvantage. A week later, returning from a coaching session with Hopkins along with Fischer and another undergraduate, Hugh Blackburn, who came from a well-to-do Scottish family, he “was maliciously glad to find that Fischer had not done all the problems. Blackburn had got solutions for all, but nobody had given interpretations except myself.” But his confidence swung up and down. “I am beginning gradually to be violent in my apprehensions regarding Fischer since we have started Mech[anics],” he wrote a couple of weeks later, but just three days after that he reported to his father that “Fischer does not get on quite so well with the statical problems, as he did in Geom[etry] of 3 Dim[ensions], and if he continues so when we come to Dynamic, I shall not be so much afraid.”

February and March of 1843 saw William, for the only time in his life, doubting the feasibility of a career in mathematics. In conversations with friends the idea came up that he might take up the law for a profession, as the Glasgow mathematician Archibald Smith eventually did. “I have pretty nearly determined to go to the Chancery bar, if something else do not succeed, though I cannot get over the idea of cutting mathematics,” he confided to his diary on February 19 (employing a now obsolete subjunctive). A month later he jotted down the same dreary thought. He attended a court hearing in Cambridge, to get a feel for the thing, and was impressed by a lawyer’s eloquent speech, less so by the somnolence of the judge.

This was a low time. He worried about Fischer beating him, agonized over his future, and from time to time was homesick and (a few months short of his 19th birthday!) nostalgic over past triumphs. A diary entry of March 14 finds him melancholy indeed. Looking over issues of the Cambridge Mathematical Journal containing his youthful work, he “spent an hour at least in recollections. I had far the most associations connected with the winter in w^h I attended the natural phil[osophy class, in Glasgow] and the summer we were in Germany. I have been thinking that my mind was more active then than it has been ever since and have been wishing most intensely that the 11th of May 1840 would return. I then commenced reading Fourier, & had the prospect of the tour in Germany before me.”

In this febrile mood, William succumbed to another temptation: literature. “Blackburn I find is a great man for reading Shakespere, besides Beaumont & Fletcher, Ben Jonson, &c,” he recorded on March 24, and a little later he sampled the dangerous fruit himself: “looked over some of Shakespere’s poems, and have just seen enough to make me wish to see more.” He even set aside his extracurricular study of the French mathematician Poisson for Richard III and Henry IV. A modest taste for Shakespeare was no cause for alarm, but Fischer was dabbling in materials far more perilous: “I went to see Fischer and found him reading [Goethe’s novel] Wilhelm Meister. I got him to read me over some in German.” Later that evening William went to bed but couldn’t sleep and got up again. “I have been looking out of the window, and have got back my journal to endeavour to fix my impressions. The moon is shining brightly on the mist w^h lies on the meadow (like the like the [sic] silvery clouds we saw from Ben Lomond. I have been looking out of the window for a long time, and listening to the distant rushing of waters, the barking of dogs, and the crowing of cocks.”

Now he was hooked. He picked up Fanny Burney’s 1778 novel Evelina, a racy, breathless tale of society and manners, and a best-seller in its day. Evelina is a young woman whose mother died giving birth to her and whose high-society father, so she believes, has disowned her. She is raised by a loving but limited country parson, who reluctantly lets her be introduced into society by better-connected acquaintances. It’s a ripping yarn, full of snobbery and disdain and gentlemen behaving badly. Inevitably, a young noble, Lord Orville, eventually marries Evelina, although not until she has survived various scrapes and perversely misunderstood his intentions.

Aspects of the tale may have struck a nerve with William Thomson. Like Evelina, he was a young provincial full of talent and promise. Where Evelina was trying to make her way in the beau monde of London and Bath, William was momentarily struggling in a Cambridge society quite different from the Glasgow world he grew up in.

More particularly, Evelina’s guardian, the Reverend Villars, may have put William in mind of his own father. Villars worries that Evelina, introduced to the brilliance and excitement of society, would develop the taste for a life ultimately forbidden to her because of her lack of means and connection. Villars observes: “A youthful mind is seldom totally free from

ambition; to curb that, is the first step to contentment, since to diminish expectation, is to increase enjoyment” and later “I would fain guide myself by a prudence which should save me the pangs of repentance.”

Evelina is caught between Villars’s cautionary admonitions and the excitement of new adventures with people of wealth and ton; she had to contend with “gentlemen” who, thinking her a naïve country wench, were eager to deceive and use her badly, as the euphemism had it. William’s predicament was hardly so dire, but as his father constantly reminded him, Cambridge abounded in wealthy, idle young men who stood ready to draw unworldly young people into loose living and depravity.

Burney’s novel is a genuine page-turner, and William’s eagerness to stay up late and find out how it all ends is easy to understand. But this tale of the innocent abroad evidently awoke unfamiliar feelings. One evening after reading for a while, “I spent a long time looking at the sheep, and listening [sic] the birds, whose singing filled the air…. I got to bed with a very strange feeling.”

The next day he had the usual round of walks and conversations with friends, then returning late picked up the book again: “On Sunday night, after I was left alone, I read Evelina till 2^h 20^m, when I finished it (the 1st novel I have read for 2 or 3 years).”

Evelina’s tale, after a series of implausible coincidences and discoveries, ends in unalloyed happiness. Her true aristocratic parentage is revealed and her fortune restored; she reconciles with her father, who had been deceived into raising another young girl as his daughter; she marries the noble Lord Orville; the salacious Sir Clement Willoughby is sent packing. But as Burney sharply relates, the revelation that Evelina is a blue blood after all causes all the previously disdainful society women to embrace her as one of their own. William Thomson could expect no such absurd denouement. If he felt at all uneasy in Cambridge, out of his depth with people whose style and manners were strange to him, unsure of his future, he could expect no miraculous lifeline. The blithe passage of his youth and the easy renown he had achieved in Glasgow paled a little before the contemplation of his new environment and the future it promised. If he were to prosper it would have to be entirely his own doing, and for a few months that burden troubled him.

The day after he had closed the book on Evelina’s thrilling adven-

tures he went out to scull for a while, came back to his rooms, answered a letter from his father, then “went to Fischer. I find he has been reading Goethe to a great extent.”

Wilhelm Meister, like Evelina, is a young person seeking a path in the world. He falls in with an itinerant set of actors but soon finds them shallow and vain and wonders where his future lies. Searching for his vocation is a solemn task for young Wilhelm: “What mortal in the world, if without inward calling he take up a trade, an art, or any mode of life, will not feel his situation miserable?” After a series of mishaps and adventures and coincidences even more implausible than Evelina’s, Wilhelm gives up his dreams of the stage and begins to see a more responsible and worthwhile future. “To thine own self be true” is the message, with the implicit assumption that one’s own true self will turn out to have merit.

Early in May, James Thomson wrote that he would be coming down to London over the summer because Robert again needed surgery to remove stones. The prospect of a visit from his father overjoyed William: “I hope most intensely that you will come here, instead of waiting in London to meet me.” What passed between father and son is unrecorded. From London at the end of May, James Thomson wrote warmly of his visit: “With my trip to Cambridge I have been much gratified. I am glad to say that what I saw and heard of you was very satisfactory. Your success in your studies, and in making the most valuable of all acquisitions—character, has afforded me great pleasure.” He was pleased with the circle of friends his son had formed and, clumsily but earnestly, offered William encouragement at rowing: “Tell me about your races. You see that though I consider it necessary you should give them up for the future, yet I feel an interest in them so far as you are concerned.”

William broke off his diary while he was in Scotland for the summer. Interpretation is difficult not only because of its sparseness but also because pieces of it have been carefully torn out—sometimes a line or two, sometimes a paragraph, sometimes whole pages. The diary came to the Cambridge physicist Joseph Larmor in this condition after its author’s death.

When William returned to Cambridge, on the portentous date of Friday the 13th of October, so too his anxieties returned. He tried his diary again, but even more sporadically than before. On Sunday he re-

corded a few casual remarks, then something more personal, of which only this remains, part of a page that was incompletely torn away:

The next day there is a little more: “During last week I have been rather unsettled and not applied myself to reading nearly as much as I could have wished. The idleness however did not depend upon external circumstances as I have been in my room almost every night at or before 7, but partly to my having Wilhelm Meister….” The rest of the page is torn off. When it resumes he mentions reading mathematics with Blackburn, and there is more about boats, but “I was very little interested about the race.”

The date of this entry is unclear. In late 1843, though he resumed his studies with a vengeance, he also kept up with rowing. At the end of November William wrote cheerily to his sister Anna that although he had to catch up on a lot of work for Hopkins “I am practising now everyday for a great skulling race w^h will take place on Tuesday. The winner will get a cup of about fifteen guineas value as well as the honour of holding a pair of silver skulls in his hands for a year. I don’t however aspire to such an honour, and I shall be very well satisfied if I come in second or third. Blackburn and I went on very regularly with Faust till James [his brother, who visited Cambridge briefly] came, but since that we have been rather interrupted. I have very seldom time now to take out my cornopean, but after the skulling is over, I mean to miss going down to the river one day at least in the week, and to have some practice on the cornopean.”

His father and sisters, learning of his renewed interest in rowing, became fearful again, and William wrote back in his usual reassuring way. Anna replied this his letter “containing all your reasons for having joined the boat races … has one good effect at least—that of convincing us all that you are a most excellent logician.” Despite his protestations he came first in his sculling race and won the Colquhoun Cup. To the end of his life he remained immensely proud of his athletic triumph. Fifty years later, when he was supposed to be replying to letters congratulating him on being raised to the peerage, he fell to reminiscing and declared that getting the cup was “better than winning in an examination.”

He ended his diary for good with an undated half page, perhaps written after his victory on the Cam. He described his latest work with Hopkins, then announced—to himself? to his father’s shade?—that from now on he would keep only a private mathematical journal. He concluded: “I need not stop to commemorate anything about boating last term on this, as I suppose I shan’t forget it in a hurry. I shall at least remember everything worth remembering about it.”

There is an echo here from Wilhelm Meister, who after giving up his pretensions for the stage, mused that he had stayed with the actors “longer than was good: on looking back upon the period I passed in their society, it seems as if I looked into an endless void; nothing of it has remained with me.” Did William feel the same away about the time he had spent reading plays and poetry and novels? If, after he closed his 1843 diary, he suffered any further lack of confidence, it stayed securely within his own mind. No one would ever remark that he was a thoughtful, reflective, introspective man. He developed no taste for deep literature. These were childish things, and by the end of that tortured year he had put them all away. A diary entry from April 12 betrays, rather comically, a sense that William didn’t have the stuff of dark yearning in his soul. After staying up late reading Shakespeare one evening he remarked: “This is a beautiful moonlight night, and my rooms are quite romantic. If I were only sentimental enough to enjoy it, I might lose a great deal of time looking out of the window.”

He went on, during the rest of his life, to fill some 150 notebooks, bound in green, with mathematical calculations, jottings and ideas on science, drafts of papers and letters, occasional philosophical reflections—but nothing personal.

A sharp mind may make its course along many channels. William Thomson’s mathematical prowess showed itself early, but at Cambridge, for a few months, as he turned 19, a different sensibility briefly awoke. By the end of 1843, however, these intrusive and unsettling feelings had been put to rest. Thereafter William worked every waking hour at science and showed only a passing and conventional interest in music or art. On the rare occasions, much later in life, when he read a novel, it would usually be a seafaring tale. He liked Beethoven, Mozart, and Weber and was especially fond of the Waldstein, one of Beethoven’s perkier piano

sonatas. He told one of his nieces off, light-heartedly, for playing Grieg on the piano while he was in the next room. Too modern! He didn’t particularly care for Wagner, not so much because of the music but on account of the silliness of the plots.

***

The following summer, 1844, William Hopkins organized an extra session of mathematical coaching at Cromer, on the East Anglian coast overlooking the North Sea. “This is a very pleasant place, for England, and especially for Norfolk, w^h is rather remarkable for its dullness,” William reported to his father. He lodged with Fischer and Blackburn in an old house near the cliff-top, which “cannot survive another winter, I think, unless great care is taken to protect the cliff below it with a wall.” Jeopardy College, they called it. The students saw Hopkins every other morning; William saw him alone the intervening mornings. They scrutinized old exam papers. In the afternoons they read or went bathing in the sea. James Thomson complained about the expense—“Your lodgings are surely unnecessarily fine. For what you pay, we could have good lodgings on the Clyde for no inconsiderable family”—but paid up anyway, even when William had to write again a couple of weeks later asking for more money.

Whether studying or enjoying seaside amusements or both, William failed to write enough letters to his siblings, bringing a rebuke from his youngest brother, Robert: “I think you might write a little oftener…. John is the only one in this house you deign to write to except Papa & that when you want money.” William was a seasoned student now. For all his father’s scolding, he knew he could rely on getting £5 or £10 to sustain him, even if he had to plead, apologize, and ask twice. He was learning his independence, studying with his undergraduate friends, becoming closer acquainted with Hopkins, making plans to travel for a while after the summer session with his friend Blackburn.

He did not neglect his mathematics. Hopkins wrote to reassure James Thomson that William was not idling. “I am happy to say that he has given me entire satisfaction. His style is very much improved, and though still perhaps somewhat too redundant for examination in which the time allowed is strictly limited, it is very excellent as exhibiting the capacious-

ness of his knowledge as well as its accuracy. I consider his place as quite certain at the tip-top…. I hope we shall be able to send him forth with such a character as few are able to carry with them from the University.”

Back in Cambridge after a short break in Glasgow, William girded himself for the final run at the wrangler competition. He had now completed three years and was in his tenth Cambridge term. Honors exams took place early in the new year, and candidates had one last chance to study and cram before the grueling test came upon them. Expectation grew. William should be an exception among senior wranglers, so great by now was his reputation. “To have him come out as a common place senior wrangler,” Hopkins had written to James Thomson from Cromer, would be “a grave disappointment.”

By the middle of December, William’s tutor, Cookson, was writing in the same vein: “Your son is going on extremely well. He says that his health & spirits are good though he is perhaps a little more pale than usual…. We fully expect him to be first and indeed it would be a great disappointment to all his friends and a great surprise to the University if he were not—I do not know of any candidate whom he has any reason to fear.”

The candidate himself was not so sure. There may have been giddiness in Glasgow, but William urged calm. He wrote to Aunt Agnes Gall, his late mother’s sister who was housekeeper for James Thomson’s family: “I do not feel at all confident about the result, but I am keeping myself as cool as possible, and I think I shall not be very much excited about it when the time comes. One thing at least I am sure of is that if I am lower than people expect me I shall not distress myself about it, and if any of you lose any money on me I shall consider it your own fault for giving odds.”

To his father a few days before the ordeal began he wrote: “The prospect is of course rather terrible, as all the three year’s [sic] course of Cambridge reading is for the one object of getting a good place, so that in that respect anyone’s whole labour may be lost very easily. I hope however that if I do not get as high as people expect, that it will not be much disappointment, as I think my time will still not have been quite thrown away…. I am sure that many others will be quite as well prepared and I am determined to be satisfied whatever may be the result, and I hope you

will not be disappointed if I do not succeed well…. You need not be in any fear about my health as I never have been better than I am now.”

***

On top of the pressure of the wrangler competition, another battle lurked in the back of William’s mind, the result of a plot cooked up by his father with William’s not wholly enthusiastic assent. William Meikleham, the Glasgow professor of natural philosophy, was old and ailing. He had been a professor since 1799 and was 70 years old when William started at Cambridge. Though not an original scientist, Meikleham had been a good teacher and, like James Thomson, a devotee of the modern French style in mathematical science. In 1839, however, his health had begun to fail, and his lectures were delivered by substitutes, first Nichol, the astronomy professor, and then David Thomson (no relation), a young Cambridge graduate. But as long as he breathed, Meikleham remained professor. The question of his successor discreetly arose, with James Thomson taking a close interest: He wanted a lucid teacher, a modernist, a scientific man in the new fashion, and from time to time he quizzed William about some of the Cambridge fellows he had encountered. Gregory was a possibility, but he died unexpectedly in 1844. Archibald Smith might serve, but he had exchanged his Trinity fellowship for a more lucrative career as a London lawyer.

As time went by and Meikleham clung to life, another possibility came to James Thomson’s mind: his own son. In 1841, when William started at Cambridge, that idea was absurd. A couple of years later, with William’s reputation rising not merely as a likely wrangler but as an increasingly prominent contributor to the Cambridge Mathematical Journal, James saw ever more clearly before him the prospect of his son as his professorial colleague. He found ways to throw out the suggestion to some of his closer Glasgow associates (including the professor of medicine, who regrettably for our story was also named William Thomson⁶).

William’s attitude toward his father’s scheming is hard to judge. James

Thomson, from his earliest moments, had plotted out his life with a view to security and income. William, raised in a comfortable household and aware of his own talents, worried less about the future. Something would turn up; he would find a way, as he always had thus far. If he did well as wrangler, a Cambridge fellowship was his. He conceived of spending some time in Paris, to make the acquaintance of the French scientists whose work and style he so admired. Like the young researcher today, he could imagine spending a few years here and there before settling into a permanent position. His father, on the other hand, knew how rarely a secure post came open. If the Glasgow chair went to another young man, it could be closed out to his son for the rest of his life. In the whole of Great Britain no more than a handful of comparable positions existed. So he kept the fire gently alive and wrote to William of the ups and downs of Meikleham’s health.

Toward the end of 1843, when William still had another 18 months at Cambridge ahead of him, James Thomson delivered worrying news: “I am sorry to say that Dr Meikleham has a second attack of his distemper, and that, though he may yet get over it, he is considered to be in a most precarious state. I wish he were spared for two or three years longer. In such things, however, we have no controll [sic].” He asked William for his opinion of Gregory or even Hopkins. The great concern at Glasgow, he explained, was that Cambridge men were thought too abstract, too mathematical, too superior. They wanted a practical man, who knew a little bench science, and could teach to a less exalted student body than attended Cambridge.

William made only passing and noncommittal responses to these overtures. Next April his father reported a conversation with Dr. William Thomson, the medical man, concerning Meikleham’s successor: “I felt … I ought to mention to him my views regarding you…. He was naturally quite struck with the idea of your youth, &c.; but he received the proposition as favourably as could be expected. He asked about your experimental acquirements, particularly in chemistry … and he said that a mere mathematician would not be able to keep up the class.” James Thomson advised his son to find a way of doing some laboratory chemistry and relayed Dr. Thomson’s opinion that although there was no doubt of William’s being “an accomplished analyst in mathematical and physi-

cal science, yet it would operate much against you … should you not be able to give evidence of your acquaintance with the manipulations, to a certain extent, of experimental philosophy. Turn the whole matter carefully in your mind, and write to me soon about it.”

William replied that he could not possibly do chemistry experiments in his college rooms and promised vaguely that he would look for some alternative. But his father, the bit between his teeth, galloped on regardless. More than ever he reminded his son of the necessity of mature and circumspect behavior: “What you have to do, therefore, is to make character, general and scientific, so as to justify the Lord Rector, the Dean, and the other electors who usually act with me, in supporting you—a matter of difficulty on account of your youth.” Warning of one of the senior Glasgow faculty members “who, as to private affairs, is more nearly omniscient that any one I have known,” he reminded his son yet again of the perilous associations of rowing: “Avoid boating parties in any degree of a disorderly character, or any thing of a similar nature; as scarcely any thing of the kind could take place, even at Cambridge, without his hearing of it.”

These events unfolded during the same spring that William fell under the influence of Evelina and Wilhelm Meister. Career planning was not uppermost in his mind. After his father’s visit to Cambridge, he reapplied himself to his studies. Jockeying for the Glasgow chair occupied his father’s attentions far more than his own. In August James Thomson reported that “Dr. Meikleham has had another attack—a very bad one. He has weathered it, and is pretty well again. In all probability, he will not survive another.”

That spring William spent a week or so in London, staying with his older brother James, who was then apprenticed to an engineering firm in Millwall. He also visited Archibald Smith, who despite taking up the law advised William to stick with science. Smith “seems to be getting on very well, and I think now has no idea of giving [the law] up, though he says that he thinks of all lives that of a professor must be most enviable,” William told his father. “He said that I should not go to the bar, and when I said that I might not be able to get anything else, he answered that if Dr. Meikleham would live a little longer, I might be appointed his successor.”

Just a couple of weeks earlier William had lamented to his father that “three years of Cambridge drilling is quite enough for anybody.” Eager to apply himself to real science rather than formal study, William finally began to see the attractions of a professorial appointment beside his father and for the first time began to write as if it were his ambition and not just his father’s dream. In April Meikleham suffered yet another setback, falling in his room and bashing his head on the fireplace. William responded to this news with less than commendable concern: “For the project we have it is certainly much to be wished that he should live till after the commencement of next session.” By the middle of May, James Thomson reported, Meikleham was “greatly changed. He is silent—vacant—and seems to notice little of what is going on around him…. I shall be much surprised indeed if his chair be not vacant before the beginning of next session.”

Against the odds, however, Meikleham held on, reduced almost to a vegetable state, clearly incapable of teaching, but professor of natural philosophy still. William studied at Cromer over the summer and returned to Cambridge for the final assault on the mathematical tripos. Meikleham continued to loom silently over his life. Dr. William Thomson kept up with advice to his younger namesake, suggesting he try writing a popular lecture on some scientific subject to prove that there was more to him than rarefied mathematics. “A Cambridge education did not always give the power of easy expression or of commanding the attention of an audience,” the older man had remarked to James Thomson, with an ironic smile.

As William readied himself for the exams, disquieting news emerged from St. John’s, one of the larger colleges and a traditional leader at wrangling. They had a student, Stephen Parkinson, who displayed an extraordinary capacity for absorbing old exam answers and blurting them out again at high speed. Parkinson crammed until he was gray, and as the examinations unfolded, the Johnians felt confident enough to place bets on their man.

On January 1, William dashed off a brief note to his father to say that he had cruised through the first two papers with time to spare but afterwards thought of things he had forgotten to put in. His tutors also wrote to keep James Thomson abreast of the drama. Hopkins reported

that William “is going through his examination with vigour and cheerfulness, and in good health, and as far as I can judge has, in the majority of his papers, done himself justice…. My confidence that he will be senior wrangler has always been very strong. and I can only say that it remains unshaken.” Cookson sent on one of the exam papers with the confident assertion that “I think that he cannot fail to do almost every question in this paper.”

But on January 5, 1845, a rest day in the middle of the examination, William wrote with mixed news: “I have been getting on very well with the examination, and certainly quite as well as I expected…. However, I hear the Johnian has been getting on exceedingly well and so I must not be too confident…. Yesterday I was told I was the only man who did not look seedy with the examination.” A week later, when the trial was over but the outcome still unknown, he reported that “the Johnians are talking confidently of their hero…. I have not been making myself anxious on the subject.”

The competition, always an exciting event on the Cambridge calendar, became unusually intense that year and was the subject of much college gossip. A firsthand account comes from Charles Arthur Bristed, an American student who wrote a memoir of his five years among the natives:

Parkinson’s style of automatic writing was not for Thomson. Despite coaching, he could not prevent his mind from running away with him. He would always see fascinating implications beyond the immediate scope of the question, or a way of solving the problem that might turn out a little cleverer than the approved way. Or not: How he could tell until he tried? He saw more than the typical examinee and was duly penalized.

When students and faculty gathered at the Senate House on January 17 to see the results posted on the notice board, it was William Thomson’s friends who were downcast. He was junior wrangler, Parkinson senior. Bristed’s tale resumes:

William’s prior hesitance now allowed him a perverse kind of victory over his father. “You see I was right in cautioning you not to be too sanguine about my place…. The only thing I feel in the least degree about it is that it may make it more difficult to succeed in getting the professorship in Glasgow…. I hope you will not think I have misspent my time here. I feel quite satisfied that I have spent as much time on reading and preparation as I could consistently with higher views in science. The Johnians give themselves up to one object, and it is fair that they should have their reward.”

He wrote again the following day: “I hope by this time you have recovered from the shock of what I am afraid you must have considered very unpleasant intelligence…. Ellis tells me, and does not hesitate to tell others of his friends, that even without previous opinion, he could see by my papers that I am better than Parkinson, but that I fell short in quantity.”

His father’s guarded reply would only partly have assuaged him: “The place you have got at the examinations is an excellent one, and you and

all of us ought to be well satisfied with it. In point of name the next higher place would have been desirable; but coupling with your place all the distinctions that you can claim, we can and will make out a good case for you.”

From his sister Elizabeth he got warmer reassurance: “I must confess that the unlooked for result of the examination has somewhat disappointed me; but Papa says he thinks you have the character of Senior Wrangler notwithstanding, and he trusts you will maintain it…. I was very sorry in reading your letter which arrived Sunday when I came to the part where you say you are afraid Papa will think you have misspent your time at Cambridge. He does no such thing, he is very proud of his son and not in the slightest degree less pleased with him since the small humiliation he has met with.”

It was James Thomson’s perennial weakness not to be wholehearted in any judgment until he had obtained authoritative support. From Hopkins he soon received a lengthy and unreserved testament to his son’s abilities: “I confess that your son’s not being senior wrangler is to me a very great disappointment. I can assure you however that the circumstance has not affected in the slightest degree the high opinion in which I hold both his talent and acquirements…. The fact of your son being second is perfectly explicable without lessening the conviction that in the high philosophic character both of insight and knowledge, he is decidedly first…. While others are simply answering a question, he will often be writing a dissertation upon it…. One of the examiners … told me he thought it highly probable that while your son would be hereafter building up for himself a European reputation, his opponent might be scarcely known beyond the bounds of the University…. Your son bears his disappointment extremely well, better I think in fact than his friends.”

This assessment was acute: while Thomson went on to build a reputation greater even than Hopkins imagined, Parkinson remained in Cambridge for the rest of his life, achieving little except the induction of further generations of diligent men into the realms of wranglerdom. James Thomson seized on Hopkins’s letter, not only to dissolve any personal disappointment but to fire up again his project of bringing William to Glasgow. He made sure the letter came to the attention of senior Glasgow faculty and joyfully told William: “Hopkins’s letter has done you great

good here. The man that wrote it has a heart and a head.” Professor Buchanan read the letter in the Thomsons’ dining room and, said James, “exclaimed ‘that is the kind of man we should have!’ I do not see any meaning to put on this except what at once occurred to myself and what will readily occur to you.”

William had one final ordeal before concluding his student career. The Smith’s prize examination came after the honors tripos and was reckoned to be a deeper test of scientific understanding than the mad race to be wrangler. Each year students tried to answer a number of long questions on mathematical physics rather than pure mathematical methods. The Smith’s prize rewarded analytical thought more than trained rapidity of response. It too had its own mystique and pressures. Shortly after Thomson’s time, the Smith’s prize examination was overseen (“invigilated,” to use the preferred Cambridge term) by his close friend Professor George Gabriel Stokes. The exam took place at Stokes’s house in Cambridge, in the dining room. Mrs. Stokes and the children took their meals in the kitchen for the few days the exam lasted, and tiptoed about while the young students huddled at the dining table, scratching their pens at the papers while Stokes, a taciturn man at the best of times, said nothing except presumably, “Turn the paper over!” and then, some hours later, “Gentlemen, put down your pens.” The imagination easily supplies Stokes’s even breathing, a clock ticking somewhere in the quiet house, hushed childish voices and padded feet beyond the firmly closed dining room door. Stokes’s daughter recorded years later an occasion when two young men, having survived the morning session, felt unable to face the afternoon and ran away through the garden after eating the lunch Mrs. Stokes had prepared. An unfortunate incident, Stokes commented ruefully, as the two men had actually been doing rather well. Thereafter he made sure the garden gate was locked before he started the exam.

In 1845, William handily beat Parkinson for the Smith’s prize. Cookson conveyed the good news to James Thomson: “I have seen your son, who is overjoyed…. Some of the papers for the Smith’s prize examination were of a more difficult nature than those [for the tripos] and required a more profound & philosophical view of the subjects. It is to this that your son’s success may be attributed.”

Overjoyed he may have been, but William wasted no time in self-

congratulation. Meikleham was hardly speaking, barely breathing, but still alive, and for the time being neither he nor his father could do anything to bring the Glasgow appointment closer. William had had his fill of reading and studying and cramming. Now it was time for his real education in science. On January 24, 1845, he wrote hurriedly to his father with the result of the Smith’s prize. The next day he went to London with his friend Hugh Blackburn, and from there the two continued to Paris, where many of the greatest exponents of mathematical and experimental science lived and worked.

***

As well as Fourier’s mildly controversial work on heat, published in 1822, the great classics of French mathematical science from the late 18th and early 19th centuries include Lagrange’s Théorie des Fonctions Analytiques (1797), Laplace’s five-volume Traité de Mécanique Celeste (1799-1825), Cauchy’s Cours d’Analyse (1821), and Poisson’s Traité de Mécanique (1833). Taken together, these monumental works created the modern form of differential and integral calculus and their application to mechanics and the motion of bodies. No matter what inscrutable processes of thought Isaac Newton had employed to devise these ideas in the first place, when it came to compiling the tremendous Philosophiae Naturalis Principia Mathematica of 1687 he reverted to a presentation that Euclid would have recognized. Far from being a system of mechanics, Newton’s work strikes the modern student as a collection of geometrical exercises and trigonometric problems. This style not only made the book hard slogging, it concealed much of the mathematical structure of the underlying theory. The French mathematicians enlarged and systematized Newtonian mechanics into the sophisticated body of analytical methods that students learn today.

French scientists had also raised the art of experimental investigation to new importance. Charles Coulomb and Jean-Baptiste Biot had established laws for forces acting between electric charges and simple magnets. André-Marie Ampère measured interactions between magnets and electric currents and proposed a sophisticated mathematical theory that held sway for a decade or two. In optics, Biot and Augustin Fresnel and Dominique-François Arago investigated the polarization and diffraction of light

and attempted to tie their findings into mathematical systems from the pens of Poisson and others.

The French achievements of this era became, and remain, the foundation of a “mechanical” description of nature, consisting ultimately of inanimate objects responding to elementary forces. What the French called la physique aimed to combine experimental investigation and mathematical sophistication into a seamless whole. In 1816 Biot published a textbook, Traité de Physique Expérimentale et Mathématique setting out what we would now call (to use a much abused word) a reductionist view of natural phenomena, in which forces acting on particles are at the bottom of every physical process. Of course, there were mathematicians and scientists outside France too: Euler and Gauss in Germany, Young and Herschel and Babbage in England, Brewster and Nichol in Scotland. But Paris was the birthplace and center for a view that came to influence the rising generation in Britain and eventually penetrated Cambridge too.

One man who grasped quickly the superiority of these gallic innovations was James Thomson. In 1825, writing in the Belfast Magazine and Literary Journal, he offered this scathing judgment: “Since the days of Newton, however, the British mathematicians have been far surpassed in several branches of science, by their neighbours on the continent [especially in] the higher and more difficult parts of pure mathematics, and in physical astronomy.” Here he noted particularly the work of Lagrange and Laplace, as well as Euler, and went on: “While these brilliant achievements were crowning the efforts of the mathematicians on the continent, the men of science in Britain were wasting their time and talents, some in restoring the ancient geometry of Greece, and some in following servilely and implicitly the manner in which Newton presented his investigations, without being actuated by the spirit by which he was directed in his researches. Fond and proud of that eminent man almost to devotion, and prejudiced against his rivals on the continent, partly by feelings of national jealousy, and partly by the scientific war between the adherents of him and of Leibnitz,⁷ they generally clung, even in the minutest particulars, to the methods pointed out by their great leader; and, falling behind

on the march of discovery, they scarcely contributed in the slightest degree, during the lapse of a century, to the advancement of science, in its higher and more difficult parts.”

But as James Thomson went on to say, change was afoot. A group of reformers led by William Whewell (who brought the words “science” and “scientist” into common parlance), William Herschel (discoverer, in 1781, of Uranus), and Charles Babbage (designer of the famous “difference engine,” a mechanical calculator) began to clear away the old dogmatic Newtonian lore in favor of the more flexible, systematic, rigorous, and general mathematics of the French school. Or rather, at Cambridge, this was half a reform: Cambridge slowly embraced French mathematics but failed to succumb fully to the charms of la physique. In an intellectual divide that remains today, at least at the level of stereotype, the French hankered after a grand overarching system in which all phenomena ultimately referred back to theories of a single, logically consistent formulation, while the pragmatic Anglo-Saxons preferred to analyze empirical matters piecemeal. The French wanted to portray mechanics, light, electricity, magnetism, and heat all as parts of a universal underlying theory of matter and forces. The British were content to come up with satisfactory accounts of each of these subjects on their own terms.

By the time William Thomson studied at Cambridge, Whewell was ironically beginning to seem like part of the old guard, resistant to further continental scientific innovations (the “despotic Whewell,” William once called him in a letter to his father, because of his resistance to change in the mathematical tripos). In Paris, William quickly became acquainted with the surviving French savants of the great generation, including Biot and Cauchy. He met Joseph Liouville, editor of the Journal de Mathématique, who set him a problem. Like his French colleagues, Liouville had difficulty with Faraday’s schematic but suggestive portrayal of the electric tension between objects as an influence carried along curved lines spreading throughout space, and thought it contradicted Coulomb’s inverse-square force acting along a straight line between two charges.

This notion—action at a distance, as it was usually called—was a piece of Newtonian thinking taken unchanged into la physique. Faraday found action at a distance philosophically objectionable and believed that there must be a medium pervading space by which electric forces transmit themselves from one place to another. His lines of forces were an attempt to capture this vision.

Full development of Faraday’s insights lay in the future. For the time being Liouville could see only one theory versus another: action at a distance along straight lines versus curving, elastic lines of force. As William reported to his father, “He asked me to write a short paper for the Institute explaining the phenomena of ordinary electricity observed by Faraday, and supposed to be objections fatal to the mathematical [i.e. French] theory. I told Liouville what I had always thought on the subject of those objections (i.e. that they are simple verifications) and as he takes a great interest in the subject he asked me to write a paper on it.” In truth, no contradiction existed. Faraday did not dispute the magnitude of the force predicted by Coulomb’s law, and for a simple case such as two charges separated by some distance, the lines of force would be perfectly symmetrical around the line joining the charges, so there would be no sideways forces. In the case of three or more bodies, William easily showed, the geometry obviously became more intricate, but Faraday’s picture nevertheless predicted the same forces as Coulomb’s law did.

But a significant conceptual difference existed between the two pictures. According to Coulomb, electrical forces acting in a complex arrangement of multiple charges were best imagined as the summation of independent forces acting between all the pairs. According to Faraday, the charges created a state of electric tension pervading the whole of space around them, and the force acting on any one charge arose from the electric tension where that charge resided. William showed that the pictures came to the same thing, however one looked at it, and for static arrangements of charges nothing more need be said. Faraday’s depiction, vague and poorly formulated as it seemed to Liouville and the French, ultimately had more physics in it. But that was not yet apparent. For the moment William had shown with simple clarity that Faraday and Coulomb did not disagree. To do this he combined hard mathematics with an appreciation of the physical phenomenon concerned. It was a kind of

problem solving at which he excelled. And he showed a sympathy for la physique while retaining a certain outsider’s perspective.

Equally important for William’s career was his stint of work in the Paris laboratory of Victor Regnault, a 35-year-old experimental scientist. At the behest of the French government, Regnault had embarked on a long project to measure the thermal properties of steam—its rate of expansion with temperature, the quantity of heat needed to raise its temperature by some amount, and so on. As the new industrial economy grew, steam power became an ever more important foundation for national prosperity. In Britain, birthplace of the practical steam engine, inventors and amateur scientists continued to develop the new technology in laissez-faire style. In France, politicians saw an opportunity to spend national revenue on a project of national importance. Efficiency in a steam engine meant economy of operation (more power from the same quantity of coal), but next to nothing was known at the time of the scientific principles behind steam power. Regnault’s work aimed to establish a foundation of practical knowledge by which to improve steam engine design.

William’s time in Regnault’s lab introduced him not only to practical science, at which he proved adept and ingenious, but also to the implications of the theory of heat in technological matters. In his reading of Fourier he had come to know heat as an element of fundamental physics. Working long days with Regnault, he discovered heat as a source of motive power, causing gases to expand, pistons to slide, and crankshafts to revolve. The French scientists, for all their love for the rigor and elegance of higher mathematics, also insisted on the importance of empirical knowledge. William’s four and a half months in Paris in 1845 turned him from an applied mathematician into a man of science. Decades later, as Lord Kelvin, he told the French Academy on receipt of an honor that France “is without doubt the Alma Mater of my scientific youth, and the source of that admiration for the beauty of Science which has enchanted and guided me throughout my career.”

***

William’s letters to his father from Paris expressed unabashed excitement at being inducted into the fellowship of true scientists. His father’s

responses were equally enthusiastic but for a less exalted reason. Meikleham ailed but lived still, and the Glasgow chair remained open. “Dr W Thomson [the medical man] and Dr Nichol are anxious that you should become acquainted as much as you can with the great men of Paris, as testimonials from them may serve you much, and it will be pleasant to make their acquaintance irrespective of this.” And later: “Dr W.T. is much pleased to hear that you have got fairly into Regnault’s Cabinet [de physique—his lab], and hopes you will be able to get a good testimonial from him … and others regarding your general knowledge of Physique, and showing that you are not merely an expert x plus y man.” James Thomson, relaying the oracular advice he regularly obtained from the older Dr. William Thomson, urged his son to get promises of a testimonial from any great Parisian he happened to meet, no matter how slight the occasion. William wrote of the wonderful things he was learning and the ideas he discussed with Regnault, Liouville, Cauchy, and the rest. At length he admitted he could probably get letters from Liouville and Regnault, but he begged off asking anyone else, saying he had had insufficient contact. He hoped those two testimonials would be adequate reward, in his father’s eyes, for his time in Paris.

He left France at the end of April with no clear plan beyond going back to Cambridge and picking up some coaching to make a living while he pursued all the new ideas he had absorbed in Paris. At the end of June, however, he wrote to his father that “very much contrary to my expectations” he had been elected to a fellowship at St. Peter’s. His protestation seems disingenuous in the extreme: as an undergraduate he was already publishing alongside the great scientists of the day, and despite his tiny failure in being only junior wrangler, one of the examiners had reportedly declared to another, “You and I are just about fit to mend his pens.”

This news brought joy to his father, tinged with wistfulness. He congratulated William on getting “forward so far at so early an age! At your age I was teaching eight hours a day at Dr Edgars,⁸ and during the extra hours—often fagged and comparatively listless—I was reading Greek and

Latin to prepare me for entering college, which I did not do till nearly two years after.”

During the summer William coached undergraduates (as he had been coached only six months earlier) and came back to Glasgow in September, catching up with his family whom he had not seen for a year. There were also, of course, “important matters in consideration at present” to be advised of by his father, and no doubt a good deal of plotting and preparation went on between them. But Meikleham, neither better nor worse, clung silently on to life. William returned to Cambridge. He had “as many pupils as I would wish” and also gave lectures in college every morning at eight o’clock, which he said he enjoyed more than the coaching.

The following February James Thomson informed his son that a teacher of mathematics at the Glasgow High School was ill and would probably soon die, and wondered if William might make a move for the position. William’s reply makes clear that he was beginning to establish his own life and would not go along with his father’s every scheme to get him back to Glasgow. He turned down the opportunity because “I am afraid I should have to give up any thing in the way of original research. At present, at Cambridge, I can with ease make more than enough money to support myself, and when I commence receiving money for my fellowship, I think with what I receive for lecturing, I should be independent of private pupils, in pecuniary respects. The only event which could make me require more than I get at present would be marriage, and if that were ever to happen when I am here, I think I could, by private pupils, get as much, or very nearly as much money as is mentioned for the situation in Glasgow, with as little work, and that of course of a higher kind.” The fellowship paid £200 a year, with rooms in college—a comfortable living.

James Thomson didn’t push the matter further and in fact asked his son’s advice on who else might be suitable. Instead, after a disturbing rumor came to his ears at Glasgow, he resumed his campaign for the big prize. He had heard, he wrote in early May, that William was “said not to bring down your instructions to the capacity of ordinary students…. Such a report may seriously injure you…. You must take care to cure the evil, if it exist; and if not, to teach so simply, clearly, & slowly, that you may be able to get decidedly good testimonials on that point. Do attend

to this above all things.” The origin of this tattle-tale never became clear. William suspected it came from a man at Trinity but in any case wrote strenuously to say it wasn’t at all true. All too cognizant of his father’s concern and tenacity, he got his former examiner, now colleague, Ellis, to write to James Thomson in the same vein: “The idea, if there is such an idea, that it is the common opinion in this university that as a private tutor he advances too rapidly or ‘talks over men’s heads’ is I verily believe, perfectly unfounded. I have never heard a syllable to that effect.” Once these fears had blossomed, however, James Thomson could not let them go, and he harped incessantly on the question of his son’s ability to teach at the appropriate level. He reported that Dr. William Thomson was now worrying that young William suffered from “timidity and want of effective locution” and wondered if he could work up some sort of nonmathematical lecture of a general nature to assuage fears that “your ideas and expressions are bound up in the icy chains of x’s and y’s, +’s and −’s.” William hardly bothered make any reply to these further charges, the possibility of calming his father seemed so hopeless.

In any case, the battle was now engaged in full. Meikleham died at last, early in May 1846, and James Thomson upped again the barrage of letters and advice and instructions to his son. He wrote with long lists of the names of eminent people who might supply testimonials. He wanted proof (what it might be was unclear) of William’s teaching abilities. He derived further anxiety from another rumor: It appeared that Archibald Smith had some thoughts of entering the competition for the Glasgow post. Smith had at first said he would write a letter on William’s behalf but then wrote a couple of days later to say he was thinking about applying himself. William visited Smith in London and found only that he hadn’t made his mind up. James Thomson, learning this, decided that some great plot or betrayal was under way. Smith wrote considerate if indecisive letters to both father and son in which little is clear except that he wasn’t sure what he planned to do.

Although Meikleham’s death was no surprise, except in coming abruptly after so long a postponement, William at first seemed uncertain how to react. A Cambridge fellowship was no bad thing. News of the ancient professor’s demise “took me quite by surprise, as of late I have been composing myself to the idea of being fixed here for two or three

years,” he wrote, but went on to say he would start arranging for testimonials. He allowed himself some reservations about his father’s all-out campaign, however: “I think it will not be a good plan to attempt to get too many [testimonials]. A few, from those who should know my qualifications such as Hopkins, Cookson [and other examiners] will I am sure have more influence without many others than if they were overwhelmed in a flood of testimonials from people who do not understand what is wanted for a professor of Natural Philosophy.”

William could never act with sufficient vigor to convince his father he was in earnest. James Thomson mentioned the Lord Rector of Glasgow University, Rutherford, and the Dean, Maconochie: “Could you ‘get at them’? Maconochie is a vain man and would be flattered by a letter from a great or learned man.” (A letter from his brother James a couple of weeks later spelled out ‘Maconochie’ at the top, in large, heavy block letters, as William was in the unfortunate habit of putting two n’s in the middle of the man’s name). More from his father: “… double your efforts to procure testimonials…. Could you have nothing from Chasles [another of the French physiciens] or Gauss? … Do all you can….” He urged him to try for Herschel and Faraday, among other notable English scientists. William had met Faraday once or twice by now, but though he offered good wishes and encouragement Faraday explained he never wrote testimonials for anyone, as he thought the whole process absurd and possibly corrupt. By mid-June James Thomson was writing, “I am afraid you are resting too quietly on your oars about testimonials…. I may add that the remark of one of your friends here on reading your note which has just arrived, is that it does not appear to be the note of a person who is trying to obtain a valuable appointment for life, and who is perhaps at a principal crisis in his career.” William’s siblings mostly stayed out of this frantic exchange, although his brother James wrote once or twice when their father was ill, to relay the latest instructions. Anna, safely away from the fray in Belfast where she was living with her new husband, William Bottomley, provided one suggestion: She told her brother to “get a beard fast so as to make you more imposing.”

James Thomson took up again the question of quantity versus quality in testimonials: “Cookson &c are right in their views about the fewness of testimonials, were all electors as philosophical and judicious as

they are themselves. [The Glasgow electors] are small men, however. Get therefore what testimonials you can from fellows and other respectable people. I wish you could get something from Herschel and Airy. Perhaps you could through some friends.” And he added the news that old Mr. Smith, Archibald’s father, had just returned from Malta, intending “you may be sure, [to] use, without much scrupulousness, every means in his power to forward his son’s views.”

This soap opera played out over the summer. James Thomson was never able to convince himself he had done enough to compensate for his son’s lackadaisical attitude. If William had been wavering about giving up the comforts of his Cambridge existence, a fortuitous intervention by the Bishop of Ely may have nudged him. At the end of June he had to go to the old cathedral city of Ely, some 15 miles northwest of Cambridge, to receive formal admission to his St. Peter’s fellowship. Colleges were still at least nominally religious institutions, and appointments needed an imprimatur from the local see. William traveled to Ely expecting to get some sort of document signed in absentia, but the bishop happened to be there and took care of business himself. William told him he had applied for the Glasgow post but would stay at Cambridge if it didn’t come through. The bishop responded, William reported to his father, with a mild reproach. He said he “hoped I do not intend to ‘fritter away my time with taking pupils here’ and spoke a good deal to the effect that it would be very desirable if men who have gone through the Cambridge course could be induced to continue studying and endeavouring to make discoveries in science.”

The result of this months-long campaign was a printed and bound volume of 29 testimonials, along with a separately printed letter from Liouville, which arrived late. Most of the notices were warm but routine, adverting to William’s great brilliance, his achievements so far and those undoubtedly yet to come. Hopkins was careful to add a sentence about William’s easy manner and amiable nature as a teacher, which led James Thomson to decide his testimonial was the best of the bunch. Cookson, the college tutor, concluded with: “He is already blessed with a reputation which veterans in science might envy, but his friends look for still greater lustre…. God grant that he may live and do honour to his country,” which rather embarrassed William.

Liouville’s remarks, transcending any chauvinism, were warm indeed: “I believe M. William Thomson is destined to attain a high rank among that stellar group [pléiade] of savants that England is justly proud to claim as her own.” He added a personal note: “Continue, Monsieur, to work as you have already done for a number of years, and brilliant success will crown your efforts…. Your future is bright, believe me.” It comes as no surprise to find that this private letter was included in the printed materials ordered up by James Thomson to bolster his son’s case.

This tortured tale ends in utter anticlimax. Archibald Smith never applied for the Glasgow chair. On September 11, 1846, the faculty met and unanimously selected William Thomson to be the new professor of natural philosophy. The relentless campaign at last over, all foes, real and imaginary, put to flight, William’s persevering father finally allowed himself the pleasure of unfettered joy. According to Elizabeth’s husband, the Reverend David King, “a countenance more expressive of delight was never witnessed. The emotion was so marked and strong that I only fear it may have done him injury.” The next day he was outwardly calmer but “every now and then a quiet, happy smile stole over his features, and he seemed quite full of enjoyment.”

William took it all in stride. Success had come again. He was 22 years old. A few weeks later, when he had returned to Glasgow, Elizabeth reported that “William does not look in the slightest degree elated. He is perfectly composed. You would hardly think that it was he who had succeeded so brilliantly.”