THE UNIVERSITY OF HEIDELBERG, in the southern Rhine valley, is the oldest in Germany. Founded in 1386, it has maintained a high reputation for most of the centuries since then, and in the 19th century it had, among other things, a selection of outstanding physicists and mathematicians. It was to Heidelberg that Boltzmann went in the summer of 1870, in his first trip beyond the confines of the Austrian academic world.
There he dropped in on a seminar being conducted by the mathematician Leo Koenigsberger, whose students were struggling to attack a problem he had set. When Koenigsberger asked the class for their ideas, a man he didn’t know, “thinner and somewhat older” than his other students, piped up from the back row. The newcomer came down to the front of the lecture room and, directly and clearly, expounded the solution to the problem. He had to compete against a certain amount of giggling from the audience, so amusing was his coarse Austrian accent to the German students.
Koenigsberger asked the stranger who he was. “Dr. Boltzmann of Vienna,” said Boltzmann firmly, as if that would suffice, and in fact it did. Koenigsberger knew the name and had heard that interesting work from this young man had already been presented to the Viennese Academy of Sciences. In describing the 26-year-old physicist who had unexpectedly showed up in his class, Koenigsberger used the German word hager, which might even be translated as “gaunt” rather than “thin.” This was the only occasion when any such word was used to describe Boltzmann; he was an enthusiast for the table, and as a young man soon became plump, then rotund. When he married some years later, his wife, Henriette, took to calling him her “sweet, fat darling.”
Boltzmann wanted Koenigsberger’s advice on a mathematical problem that had arisen in his recent research. Koenigsberger, impressed by the young man, was only too glad to oblige. But when the two were talking later that afternoon, he asked if Boltzmann had yet gone to see the university’s preeminent physicist, Gustav Kirchhoff. Hesitatingly, Boltzmann allowed that he hadn’t, and after pressing him a little, Koenigsberger found out why: Boltzmann had discovered an error in Kirchhoff’s most recent work, and as much as he wanted to meet the great man, he didn’t know whether or how he should approach him. Boltzmann had been in Heidelberg several weeks by this time, failing to overcome his diffidence.
Koenigsberger encouraged the young man to introduce himself to Kirchhoff and to find a way to bring up this delicate matter. Emboldened, Boltzmann left. A few hours later Kirchhoff came to Koenigsberger with an alarming tale to tell. He had been sitting in his office when an abrupt and unruly visitor barged in and, after the barest of introductions, blurted out, “Herr Professor, you have made a mistake!” Kirchhoff, a kindly man but accustomed to rather more deliberate German manners, was taken aback and more than a little suspicious. He wondered briefly if his visitor was deranged in some way. Nevertheless, Boltzmann was able to explain himself, and Kirchhoff realized this awkward young man was right: there was a mistake. In the end, as they got down to discussing their common interests in physics, the conversation proceeded more warmly.
Boltzmann is not known to have had any direct contact with Kirchhoff beyond this single encounter. Years later, delivering an appreciation of the older man’s life and work, Boltzmann betrayed no awareness that the oddity of that meeting was in any way a reflection of his own clumsiness. Rather, he explained that although Kirchhoff was by nature “helpful and kindly . . . it took a little effort to warm him up, which was most quickly done through scientific discourse. Then he was enchanting; he stated his opinions without reservation. . . .”
In that first meeting, Boltzmann was caught between two desires. He wanted to spread his name around and meet the outstanding physicists of the day, doing which entailed a certain humility. At the same time, he was eager for the world to know that he had found an error in the work of a senior figure and was punctilious about establishing his priority in such matters.
Through Stefan’s influence, Boltzmann was steered into a face-saving subterfuge. By the time he talked to Kirchhoff, he had already prepared a short note for the Proceedings of the Viennese Academy of Sciences explaining the other physicist’s error and his correction of it, and had written to Stefan, in the latter’s capacity as a senior academy member, asking for quick publication so as to ensure his priority. It apparently never crossed Boltzmann’s mind that rushing into print to point out someone else’s mistake might not be seen as a commendable way to gain scientific kudos, or that there might be a more diplomatic and at the same time more constructive solution. Scientific etiquette would have led Boltzmann to tell Kirchhoff privately of his error, at which Kirchhoff could have written his own note of correction, either with Boltzmann or prominently thanking Boltzmann for his helpful remark. Either way, the science stands corrected, Boltzmann’s name is mentioned, and Boltzmann gains an influential ally. Such subtleties, however, invariably eluded Boltzmann.
Nevertheless, he was persuaded to add a small postscript to his published note saying that when he had brought the matter to Kirchhoff’s attention, “he let me know he had already noticed it.” Privately, to Stefan, he grumbled that this was not actually true, but who was he to say otherwise?
At this early stage of his career, Stefan’s guidance was invaluable to the young physicist. By the time he made his trip to Heidelberg, Boltzmann was no longer based in Vienna, but had recently been appointed a professor of physics at the University of Graz, the same place where Ernst Mach had gone a few years earlier to teach mathematics. Stefan’s powerful recommendation had eased his way; he described Boltzmann as a young and promising scientist “who within a short space of time has published a series of works in mathematical physics which give eloquent evidence of his acuity, his assured mathematical knowledge—a man whose extraordinary talent I have been granted the most ample opportunity to admire.” Such an endorsement, coming from the head of the Institute of Physics in Austria’s capital city, must have opened many a door in the Ministry of Education and Culture, within which all university appointments were decided.
The move to Graz was a good appointment for so young a researcher, but Boltzmann’s growing reputation also made him something of a catch for Graz. The university there was old and respectable, but not especially distinguished in the sciences. In 1863, a new medical faculty had been established, and along with it arose a need for improved teaching of science to the incoming medical students. Physics was at the time in the hands of the 62-year-old Karl Hummel, who was distinctly not up to date with recent developments. The university had succeeded in hiring a young physics instructor from Vienna called Viktor von Lang, but he lasted only a year before returning to a better position in Vienna in 1865. The services of Ernst Mach were then engaged for a couple of years, but in 1867, after losing the struggle for the Vienna position that went to Josef Stefan, Mach left to become a professor in Prague, the capital of Bohemia, an important city within the Austro-Hungarian Empire, and home to an old and renowned university.
By this time, the elderly Hummel had been persuaded into retirement, and the University of Graz was able to hire a much livelier head of physics in the person of August Toepler, who had studied in Berlin and was then in his early thirties. Toepler, looking for a junior physicist to act as his assistant, cast his eye across recent graduates from Vienna and, with Stefan’s strong encouragement, settled on Boltzmann. With his mother and sister, Boltzmann moved to Graz in September of 1869 to find a tiny and dilapidated physics facility consisting of three small rooms above a lecture hall in what was a converted priest’s residence. One room was for a technical assistant, one served as a laboratory, and the third and tiniest was a “chemical kitchen” for experimental preparations. There was little room to do experiments, and little equipment to do them with, but Toepler succeeded in obtaining money to buy new apparatus and hire a lab assistant. He also lent Boltzmann a fur coat—Toepler had been a professor in Riga, on what was then the Baltic coast of Prussia, before coming to Graz—so the younger man could continue his experimental work during the winter in the unheated lab of the Graz physics facility.
In Graz the young Boltzmann had to plan and deliver lectures in elementary physics, which he did with adequate diligence but no marked enthusiasm. With much more eagerness, he applied in the spring of 1870 for permission to spend the summer semester traveling and thus visited Koenigsberger and Kirchhoff in Heidelberg. Here again Stefan’s help was invaluable: he used his influence in Vienna to find financial support for Boltzmann’s travels.
From Heidelberg Boltzmann went on to Berlin, another great center of German physics, but after arriving on July 5, his plans were waylaid by the outbreak of the Franco-Prussian War, the latest installment of a series of skirmishes and boundary rearrangements that had been plaguing central Europe since the revolutionary year of 1848. Through the middle years of the 19th century, Austria was militarily weak and economically straitened, and its young emperor, Franz-Josef, found himself burdened with the perpetual task of trying to keep his ramshackle realm from falling apart. He had already lost his northern Italian possessions, with the exception of Venetia, and in 1866 had been defeated by Prussia to the north. Austria and Prussia were the two dominant German powers, but neither wanted to be co-opted into a formal German nation, Austria because it had vast non-German holdings and Prussia because it aimed for preeminence rather than incorporation. In 1864, these two powers had been allies against Denmark in a dispute over the northern states of Schleswig and Holstein, but once the Danes had backed down, Austria and Prussia became uncomfortable co-occupiers of the region. Otto von Bismarck, Prussia’s prime minister, maneuvered the far less wily Franz-Josef into a war that Austria lost resoundingly. The Prussian generals wanted to take over Austria completely and occupy Vienna, but now Bismarck put on the brakes: he wanted a reduced but still powerful Austria as a counterweight to the other German states as well as to Italy, France, and Russia. Austria was nonetheless chastened by defeat, and from that point on, Prussia grew stronger and Austria weaker.
The Franco-Prussian War of 1870 was the final move in Bismarck’s chess game. This time using a dispute over the succession to the Spanish throne, he goaded France into a conflict in which the southern German states were obliged to take Prussia’s side. Austria, as well as Britain and Russia, stayed out of the way. Prussia came out on top, and in the next few years the foundations of modern Germany appeared in the form of a federation with Berlin clearly at the helm. Austria, moreover, was henceforth firmly excluded from the nascent Germany, and was left instead to deal with its own fractious provinces.
Austria’s internal weakness had already forced Franz-Josef to accept the institution of the dual monarchy, by which he was emperor of Austria and king of Hungary, presiding over separate parliaments in Vienna and Budapest. Further concessions of power to the Hungarians as well as to the Czechs and others were to follow.
By 1870, the shape of central Europe was essentially fixed for the next few decades. These wars between monarchs and their ministers and generals had for the most part little effect on the lives of ordinary people, and once the fighting was over and a treaty worked out, life quickly resumed its familiar pattern. Just 18 months after his abortive trip to Berlin, Boltzmann was back there again, keen to expand the circle of his scientific contacts and in particular to get acquainted with the rather imperious leader of German physics, Hermann von Helmholtz. The son of a schoolteacher, Helmholtz was fascinated by physics when young, but he studied medicine because he could get financial assistance in return for serving eight years as an army surgeon. He kept up with physics in the meantime through his own efforts, and in 1847, while still in military service, he published a groundbreaking piece of work that set down as no one had done before a systematic mathematical treatment of the conservation of energy. The idea that energy could be neither created nor destroyed was not new, having first been proposed in a recognizably modern form in 1841; the key to seeing the universality of the principle was the recognition that heat was itself a form of energy, not a distinct substance. Rumford’s observations in the late 1700s of drill bits boring into cannon metal had hinted that the production of heat was intimately connected to the expenditure of mechanical energy in turning the drill, but it was some decades still before the connection became both unarguable and quantifiable. Helmholtz’s work in 1847 in essence tied up all the loose ends and made the principle of energy conservation into the inviolable law it is now known to be.
Thereafter Helmholtz was able to devote his life to research. His interest in music and knowledge of physiology led to him to some notable advances in the science of acoustics and sound perception. His versatility and his personal determination made him into an energetic scientific leader of the German physics community, and by the time he was installed as a leading professor in Berlin, he was well on his way to becoming the “Reichschancellor” of German physics. Michael Pupin, an American physicist of Serbian origin who visited Berlin in the mid-1880s, portrayed Helmholtz as an imposing man, with a large head supported by a muscular neck, but with incongruously small hands and feet and a surprisingly delicate voice. Pupin was introduced to the great man by a subordinate who “bowed before his master as if he wished to touch the ground with his forehead.” Helmholtz was formal and punctilious, not easily approached by his students or colleagues.
To Boltzmann, raised in the convivial atmosphere of Stefan’s Erdbergstrasse institute, this was all rather strange and forbidding. His admiration for Helmholtz’s scientific prowess and his eagerness to make his acquaintance were tempered by the difficulty of doing so. From Berlin in January 1872, Boltzmann wrote to his mother that he had succeeded in having an interesting exchange with Helmholtz, which was especially valuable since Helmholtz is “not so accessible. . . . Although he works near me in the lab, I haven’t spoken with him very much.”
Helmholtz appeared to be the stereotypical Prussian indeed, immune if not positively hostile to Boltzmann’s casual Viennese ways. On one occasion during this visit, Boltzmann’s manner earned him a withering look from Helmholtz, which was interpreted to him by one of the junior scientists there: “You are in Berlin now,” he was informed.
On the other hand, Helmholtz quickly grasped the aim and importance of Boltzmann’s work. While in Berlin, Boltzmann presented some of his ideas on the kinetic theory of gases to a meeting of the German Physical Society and got into a lively discussion with Helmholtz afterward. It didn’t appear to Boltzmann that anyone else had much idea of what he was talking about.
Understandably so, perhaps. By this time, Boltzmann was taking kinetic theory into uncharted waters and was close to one of the greatest theoretical achievements of the era of classical physics. Although he had provided, a few years earlier, some physical justification for the correctness of Maxwell’s formula for the distribution of atomic velocities in a gas, Boltzmann was keenly aware of what was still missing. There was as yet no argument to say how and why a collection of atoms, banging endlessly into each other, this way and that, should come to obey the Maxwell-Boltzmann formula, or whether, having reached that distribution, it would stay that way indefinitely.
Because they collide so frequently, the speeds and directions of individual atoms are ever changing. An atom that’s going faster than the average at one moment might smash into another atom and find itself suddenly moving much slower than the average. Any mathematical formula that purports to describe the overall distribution of atomic velocities in a stable way must clearly be some sort of average. It must describe, at any given moment, the typical number of atoms moving at any given speed, but it cannot hope to be a complete listing of the exact state of motion of every single atom.
Here was the problem that Boltzmann recognized: How is it that atoms in a constant chaos of motion, forever crashing into each other, speeding up, slowing down, changing direction, all in seemingly unpredictable ways, nevertheless maintain an average distribution of velocities that follows a simple, invariable formula—the Maxwell-Boltzmann distribution? How do randomness and unpredictability on the scale of individual atoms give rise to order in bulk?
With the help of Loschmidt’s estimate of atomic size, Boltzmann knew that even a small volume of gas must contain trillions upon trillions of atoms. Attempting to follow the motion of every one, keeping track of every collision, every change of speed and direction, was clearly an impossible task. To make any further progress with kinetic theory, Boltzmann had to employ not simply a good deal of mathematical sophistication, but also a certain amount of brute force, along with a powerful belief that somehow an answer must be possible.
Boltzmann’s attack on this problem further revealed his capacity to discern the essential physics governing a complex problem and to use that understanding as a means of forging through to the solution. It was one thing, as Maxwell had done, to rely on a sense of mathematical consistency and elegance in order to devise a simple formula for the distribution, one that seemed to be right, appeared to behave in a reasonable way, but that lacked a true foundation in physical theory. Boltzmann, by contrast, was mathematically sophisticated but not necessarily stylish. The important thing was to figure out the answer. Many of his students from his later years recalled a favorite phrase: “Elegance,” Boltzmann liked to tell them, “is for the tailor and the shoemaker.”
In analyzing the full problem of atomic motions and collisions, Boltzmann’s mathematical power gave him the strength to plunge ahead. He did not want to take any short cuts. Picturing atoms as tiny hard spheres, he wanted to apply Newton’s laws of mechanics to their behavior and try to figure out from these elementary principles how a vast assembly of atoms, too numerous to be counted, would disport themselves. The problem seemed overwhelmingly—almost ludicrously—intractable, but he was confident it could somehow be solved. He was sure that atoms existed, and sure that they were obedient to the laws of mechanics. Somehow, in nature, hordes of teeming atoms contrived to behave in orderly and predictable ways. The process had to be understandable.
Achieving that understanding, however, demanded every ounce of Boltzmann’s intellectual fortitude. He was pushing into a dark and tangled jungle of mathematics, forging painfully and slowly on, sustained only by a belief that somewhere ahead lay a mountain peak from which the whole landscape could be surveyed.
Take a volume of gas and freeze it for an instant in time. Every atom will be caught in some precise state of motion. In principle, Boltzmann saw, he could write down a mathematical catalog of the atoms, allotting to each one its particular speed and direction of motion at that time.
Now unfreeze the gas and let time move on again; the atoms start crashing around and into each other. A moment later, this atom is moving faster, that one is moving slower, and all are moving in different directions. The catalog of atomic speeds and directions has to be drawn up all over again, from one instant to the next.
The daunting task that Boltzmann set himself was to see if he could find a way of tracking how this detailed catalog of atomic motions changes from moment to moment, in order then to see what regularity emerged. He couldn’t hope, of course, to track the catalog with absolute fidelity, since that would mean literally keeping an account of every atom and every collision between atoms. Rather, he would begin with a statistical formulation of the state of the atoms, allowing a completely general specification of how many atoms have velocities in a certain small range and in a certain narrow bundle of directions. Then he would estimate how many collisions between atoms coming at each other with a certain relative speed and at a certain angle would occur in a small interval of time. From that he could calculate how each such class of collisions would alter the motion of the atoms involved. And having done all that, he would finally perform a grand averaging over the whole complicated mess to see how the totality of all possible collisions would change the overall distribution of atomic speeds and velocities.
In taking on this fearsome task, Boltzmann had to combine straightforward and uncontroversial analyses of the mechanics of collisions with much more novel elements of statistical theory. This marked a turning point in the development of theoretical physics and in the understanding of the processes that theoretical physicists were striving to understand. As a discipline of mathematics, the study of probability and statistics had some history, going back to the efforts of Blaise Pascal in the first half of the 17th century to work out the odds governing various dice and card games. But such ideas had remained the province strictly of mathematicians. Laws of probability were all very well for games of chance, but they could not be laws of physics.
The very definition of a law in physics seemed to demand certainty, not probability. Boltzmann himself did not at first appreciate the magnitude of the revolution he was about to instigate. He knew that any particular set of atoms or molecules must have, at any particular time, a certain set of motions; the problem was that no one could possibly know what those motions were with any exactitude. His use of methods from statistical theory was, so he thought, simply a mathematical technique to help him solve a hard problem. It was not that there was anything intrinsically statistical about the motions of the atoms themselves, rather that he had to use statistics to describe atoms because of their enormous number and the almost impossible complexity of their behavior.
But even in this limited sense, statistics and probability were foreign subjects for most physicists of the time. The idea of a mathematical function representing not the actual specific state of a lot of atoms but rather the likelihood of their being in this or that state was a strange and slippery notion. The idea, furthermore, of trying to understand the effect of statistically averaged collisions on this statistical description was, to many physicists of the time, outlandish to the point of incomprehensibility.
Nevertheless, Boltzmann pressed ahead. In a 100-page paper published in Vienna in 1872, with the unrevealing title “Further Studies of the Thermal Equilibrium of Gas Molecules,” Boltzmann laid out in minute detail his analysis of atomic velocity distributions, and established a series of profoundly important results. The bulk of the paper was taken up with the establishment of an equation—Boltzmann’s transport equation, as it became known—which embodied the variation of the distribution of atomic velocities due to collisions. Through pages of laborious but purposeful calculation, Boltzmann established how a typical or average set of collisions would transform the distribution. The only gross simplification he could allow himself was to assume that the atoms are moving in random directions, an entirely reasonable assumption since the volume of gas in question is not, as a whole, moving anywhere. He established, in the end, a surprisingly simple differential equation.
That done, he had to work out what might be the solutions to this equation. A general solution was impossible; that would be tantamount to understanding in complete and microscopic detail the behavior of any random set of atomic velocities. But Boltzmann was interested in a more specific case: thermal equilibrium, as it is called. Physicists had long understood that a volume of gas at a fixed temperature will exert a predictable pressure, and that if the gas is left alone it will remain in that fixed state indefinitely. Compress the gas and its temperature and pressure will go up in an orderly manner, adjusting to the new volume by establishing itself in a new equilibrium state.
The defining characteristic of equilibrium, from Boltzmann’s new perspective, was that even though individual atomic velocities are constantly changing, the overall distribution is not. As many atoms are boosted to higher velocities as are demoted to lower ones. Boltzmann sought out a solution to his new equation that would correspond to such a state. It was, after all the preceding work, rather easy to find. There was one and only one unchanging or “stationary” solution and it was, to Boltzmann’s immense satisfaction, none other than the Maxwell-Boltzmann formula.
Boltzmann had now proved that the distribution he and Maxwell had arrived at through a mixture of guesswork and arguments from plausibility was not just the right one but indeed the only possible one. Finally this was a proof, starting from nothing but Newton’s laws for the collision of atoms, that a state of thermal equilibrium must correspond to a Maxwell-Boltzmann velocity distribution, and that the Maxwell-Boltzmann velocity distribution was the only one corresponding to thermal equilibrium.
Boltzmann’s great work of 1872 signified the arrival of a true genius of physics. His earlier work had been notable but, as is often the case in science, it was work that any one of a number of leading physicists could equally well have produced. In his 1872 publication, however, Boltzmann bulldozed a path through thickets of reasoning and intricacies of mathematics that no one else would even have dared to tackle. He succeeding in extracting a powerful equation and a simple answer where, at first sight, it was hard to see how to make any progress at all.
That, at least, is how Boltzmann’s achievement appears now. At the time, few physicists were capable of understanding his aims and methods, and fewer still had the tenacity to work through his pages of calculation. In his scientific work, as in his conversation and his personal letters, Boltzmann set down his ideas more or less as they came to him. It was not his habit to refine his writings in order to make the flow of logic clearer to the uninitiated. It apparently never occurred to him that this would be a useful thing to do, either for his readers or, indirectly, for his own benefit, in that his readers might more easily see what he was about.
Clausius, the acknowledged originator of modern kinetic theory, lacked the mathematical acumen to follow his younger colleague. Even after Maxwell had introduced the idea of a distribution of atomic velocities, Clausius never abandoned his earlier practice of treating all the molecules in a gas as if they had the same average speed. If that step was beyond him, surely Clausius could not follow Boltzmann’s further elaboration of analyzing how the distribution itself changed with time. And no one else in the German physics world had anything close to Boltzmann’s intensity of interest in kinetic theory.
Across the English channel, Boltzmann’s work found a more alert audience. Sir William Thomson, late in 1875, found himself musing on the Austrian’s arguments during a train journey, and afterward jotted a note to a colleague: “it is very important . . . the more I thought of it yesterday in the train, the surer I felt of its truth.” Maxwell, too, was following Boltzmann’s achievements, but his thoughts on kinetic theory had begun to take a slightly different tack, and he found himself unable to fully accept all of Boltzmann’s conclusions. It appears that even he, who among all physicists was capable of understanding the argument leading up to the new theorem, never pushed himself to make the effort. Word began to get around that Boltzmann had done something remarkable, but hardly anyone seemed to understand quite what it was, and it was only a few years later, when objections and counterarguments surfaced, that even a moderate number of physicists set themselves the task of trying to follow Boltzmann’s derivations.
Boltzmann had in fact achieved not only a profound result in physics, but one that brought into the world a new style of reasoning. He had used an essentially statistical analysis to establish an absolute truth, the correctness of the Maxwell-Boltzmann formula. But the revolutionary nature of this argument was not fully apparent at the time, even to its author. Boltzmann believed he had solved the essential problem of kinetic theory, and, finding little reaction from his peers, wrote nothing more on the subject for a number of years.
Boltzmann’s life at this time was simple and comfortable. He worked at his physics, and he lived in modest accommodations with his mother and his sister Hedwig, four years his junior. He gave lectures, favoring applied mathematics and the more mathematical areas of physics, especially, of course, the mechanical theory of heat to which he had already contributed so much. August Toepler provided money for some experiments, improved the state of the physics facility in Graz, and generally took care of any administrative matters. Although he was fond of weekend rambles in the countryside with his mother and sister and with some of his university colleagues, Boltzmann was a young man whose energies and thoughts were rarely distracted from physics.
One distraction had shown up, however, in the person of Henriette von Aigentler, a young student at the teacher training college in Graz. Henriette knew Boltzmann’s sister through the local teacher training institute, and in May 1873 she met Boltzmann on a school outing. Fräulein von Aigentler, then 19 years old, was an intelligent and determined woman. The year before she met Boltzmann, she had decided she wanted to sit in on science courses at the university, even though women at that time were unable to take degrees. It was generally the opinion that the presence of women in class would distract the male students, and that in any case the female intellect was insufficiently rational for the appreciation of chemistry, mathematics, and physics. This was the attitude of Boltzmann’s boss, August Toepler, who refused to let Henriette into the classroom. But she did not give in. She obtained references from lecturers whose classes she had visited testifying to her quiet and respectful demeanor, thus managing to win the approval of the university’s officials. In the winter semester of 1872 she started attending science classes, although she had to fight constantly to obtain renewed permission to sit in on lectures.
What contact there may have been between Henriette and Boltzmann during the summer of 1873 is unclear. In August of that year, however, Boltzmann left Graz to return to Vienna, where he had been appointed a junior professor of mathematics. Teaching mathematics was not his ideal choice, but the attractions of Vienna overcame any doubts on his part. Nor, evidently, had the attentions of Henriette von Aigentler impinged on him enough to cause him any hesitation in deciding to leave behind Graz and its middling university.
But Henriette did not abandon the connection she had made. In October she wrote her first letter to Boltzmann, asking for advice on her studies. She apologized for burdening him with her questions, but explained that since her father was no longer alive and her mother knew nothing of these matters, she had no one else to turn to; in any case, she added, Boltzmann’s sister had assured her that he would treat her requests sympathetically. Boltzmann’s replies to Henriette’s first letters do not survive, but she was evidently thankful and continued to write.
Her third letter, written in March of the following year, takes a serious turn. She let Boltzmann know that her mother had taken sick the previous Christmas and had died on December 30. Henriette, just turned 20, and the youngest of three daughters, was now an orphan. But she was continuing to study and again asked Boltzmann for advice and help. Fortunately she had connections in Graz, since her father had been a civil servant of some standing, and she ended up living with the family of the mayor of Graz. Still, her future must have seemed unsettled to her, and she began to turn to Boltzmann with increasing frequency and insistence. She wrote to him in April 1874 with news of her studies, and then again in June to congratulate Boltzmann on his election as a corresponding member of the Viennese Academy of Sciences, which she had read about in the newspaper.
Still there is no record of Boltzmann’s response. If he was aware of what it meant to be receiving increasingly insistent letters from an attractive young woman 10 years his junior, he showed no sign of it, or perhaps simply had no idea what was expected from his side of the exchange. But Henriette showed as much persistence in pursuing the physicist as she had already shown in obtaining permission to attend physics and other classes. In November 1874, Boltzmann wrote to her from Vienna—the first surviving record of his side of the correspondence. It is a short but sympathetic note, commiserating with Henriette over yet another death in her family, this time of her married sister.
Now Henriette was able to make her next move. She wrote again in December, and after giving more news of her studies made a request: “there is something else close to my heart. I have wanted to ask you about it for a long time, but didn’t dare. I would so much like a memento of you, namely your photograph. Really, can you send me one? It would be an object of my most sincere admiration as long as I live. Hoping for my wish to be granted. . . .”
Still Boltzmann was slow on the uptake. When he failed to comply promptly, Henriette sent another earnest and entreating letter, repeating in no uncertain terms her request for a photograph: “it’s admittedly a presumptuous request, but if you knew how much I wanted it, perhaps you would not take so long.” Finally Boltzmann sent a picture, with an apology that he had been unwell for a time. Henriette immediately sent a photograph of herself in return, and wrote shortly after to say how often she looked at her picture of him. She now kept up a steady stream of chatty letters to Boltzmann, and to his often tardy or perfunctory replies she was quick to respond with pleading and worried notes, hoping he is well, and not upset by her insistence.
It was now the summer of 1875. In July Boltzmann wrote to say he would be visiting Graz in September to attend a scientific conference and hoped they could meet. Back in Vienna again after the conference, he wrote to Henriette on September 27, asking her to marry him. His letter of proposal was serious and considerate, if not exactly passionate. He began by declaring that she had made a deep impression on him from their first meeting, and that as he got to know her better, he found in her those qualities that seemed to him most apt to underpin “a lasting sympathy between us.” He goes on: “it seems to me that lasting love cannot exist if a wife has no understanding, no enthusiasm for her husband’s striving, and is merely his housekeeper, not a comrade in a shared endeavor. Understand by this my confession that I love you.”
Henriette’s response has been lost, but it was rapid and positive. Now they wrote to each other constantly. Between the proposal and their marriage the following July, over 100 letters and a handful of postcards flew between Graz and Vienna. The correspondence is voluminous but, as to the personalities of the participants, oddly unrevealing. Along with gossip about their daily lives, there are extravagant protestations of love and expressions of anguish that the two won’t see each other again for a number of days. Both writers draw little hearts at the end of their letters: “these hearts bring you my hottest kisses.” But from neither party is there introspection or soul-searching about the nature of their love. With the proposition of marriage offered and accepted, both Boltzmann and Henriette seem mainly concerned with sorting out the immediate logistics of their lives, not their grandest ambitions or hopes.
While all this was developing, Boltzmann’s position in Vienna was not turning out as he might have hoped. The Institute of Physics was no longer in the Erdbergstrasse house Boltzmann so fondly remembered, but had moved to a new building, a converted apartment house in Türkenstrasse. He in any case had little time for physics, because of his duties as a lecturer in mathematics, which, as he had guessed, turned out not to suit him very much. He was mathematically adept—as his fundamental proof of the Maxwell-Boltzmann distribution had recently and amply demonstrated—but he was by no means a mathematician. The distinction may need some explanation. Nowadays it can easily seem that theoretical physics has become as much mathematics as physics. A page from a physics journal may look as abstruse and intimidating to the unenlightened observer as a page from a journal of mathematics. But there is a broad difference. Physicists, for the most part, take up mathematical ideas that have been developed by others, and adapt them to make physical models. They do not generally invent the mathematics they use.
Newton is a singular exception to this rule. In order to figure out how planets would revolve around the sun if controlled by an inverse square law of gravity, he had to come up with a new kind of mathematics called the calculus, and for this he is revered as a great mathematician as well as a great physicist. But he is really the only such person to be so regarded. Einstein, for example, introduced the mathematics of curved spaces into physics, but he took what he needed from mathematicians who had developed non-Euclidean geometry during the second half of the 19th century.
Theoretical physicists, even the great ones, tend to take mathematics as a set of tools and don’t spend too much time worrying about where mathematics comes from, or why and how it all fits together. Those abstract questions are the mathematician’s responsibility.
More mundanely, as Boltzmann himself well realized, there were large areas of mathematics that he knew next to nothing about. He was well able to teach a class on differential equations or on the theory of statistics and probability that he had made such good use of. But he was in no position to teach a class on elementary number theory, for example, which concerns itself with such things as the properties of prime numbers and the difference between the rationals and the transcendentals. Boltzmann had from the outset had some hesitation about his suitability for the mathematics position that had opened up in Vienna.
Still, Graz had a respectable but not particularly distinguished university, while Vienna was Vienna, the apex of the Austrian academic world. Boltzmann’s teacher and mentor Josef Stefan was keen to bring his brilliant young student back to the capital, and the elderly mathematics professor whose retirement made the new position available himself expressed enthusiasm for Boltzmann’s mathematical abilities.
Boltzmann’s doubts about his new job soon proved well-founded. Fortunately, his duties as a young professor of mathematics were only nebulously described, a circumstance he took advantage of by teaching classes in applied mathematics with, naturally, a particular attention to the mechanical theory of heat and the kinetic theory of gases. He cajoled the university into giving him some money to continue a little experimental work in physics, and at the same time traveled frequently back to Graz to work on experiments there, with Toepler, as well as to visit Henriette. During his sojourn in Vienna as a mathematics professor, following his monumental proof of the Maxwell-Boltzmann distribution, he published little in mathematical physics and devoted most of his energies to his interest in experimental physics, particularly in the measurement of electrical behavior to test its conformity to Maxwell’s electromagnetic theory. He continued to publish at a prodigious rate—some dozen scientific papers in three years—but it was only toward the end of this time that theoretical interests began once again to engage his attention.
In the meantime, the young physicist took the opportunity to learn something of the game of academic career advancement. In early 1875, the prestigious Polytechnic Institute in Zurich, Switzerland, made him an attractive offer. Boltzmann was interested, but despite his reservations about his duties in Vienna, he didn’t really want to leave. Nevertheless, he dangled the Zurich offer before the Austrian ministry and was able to obtain for himself a substantial pay raise, more money to do physics, and on top of that a written undertaking that should the university hire another lecturer in mathematics, he would be free to shift his interests more overtly toward research and teaching in physics—all this while remaining, nevertheless, a professor of mathematics.
He even tried the same trick again later that same year, when he was approached by the University of Freiburg in southern Germany. Academically, Freiburg could not compare to either Vienna or Zurich. On the other hand, Boltzmann could be director of the physics institute there, and, as Henriette wrote to Boltzmann, Freiburg might be a cheaper place to live and was in an attractive setting. Its being a small town, she added, “is an advantage for our personal life, because the conveniences of a large city have no value for us.”
In the end, however, Freiburg could not come up with an offer that Boltzmann (taking Henriette’s opinions into account) found sufficiently generous, and the Austrian ministry, having given so much to Boltzmann earlier that year, was not inclined to open its pockets again. He remained in Vienna.
But not happily. The task of teaching mathematics soon became, as he had suspected it might, more work than he cared for. He kept up a regular correspondence with Toepler, back in Graz, but his letters generally contained little more than chit-chat about physics and gossip about people they both knew. His occasional letters to Helmholtz, on the other hand, took on at times a surprisingly confessional tone, in the light of his admitted difficulty in talking to the senior physicist during his few weeks in Berlin. In between requests for technical information and advice, Boltzmann let on that he didn’t find lecturing in mathematics all that congenial, that his teaching was taking away from the time he could spend on physics, and that he would rather be a professor of physics except that no suitable position was then available. In one letter, he even told Helmholtz that his salary was barely adequate for life in “so tremendously expensive” a city, and that there were times when he wanted to live “not as a physicist but a little as an ordinary human being.”
These seemingly artless revelations may have sprung from an ulterior motive. Helmholtz was an enormously influential figure, Berlin was an important center of science, and Boltzmann may have figured it would do no harm to keep Helmholtz apprised of his unhappiness in Vienna in case anything should turn up. Unfortunately, Helmholtz’s replies to Boltzmann have not survived; Boltzmann later expressed regret that he had not kept them, but as things turned out there came a time when he had reason not to want these reminders around him.
It was not too long, in any case, before a chance to get away from Vienna presented itself. Back in Graz, his mentor Toepler was wearying of the task of running the physics department, and on top of that had broken a rib falling down an elevator shaft in the dilapidated old physics building there. At the same time he was being tempted by an attractive offer from the university in Dresden, which, after some vascillation, he accepted.
This opened up a senior position in Graz, for which Boltzmann was clearly a leading contender. It was explicitly a job for a physicist, and strictly speaking, for an experimental physicist. Whereas a ministerial document recommending Boltzmann’s appointment in Vienna three years earlier had emphasized his mathematical astuteness, a similar recommendation for the Graz position now emphasized how productive he had been in the laboratory. This was all true, to an extent, although his experimental findings never achieved the greatness of his innovations in theory. By this time, moreover (he was now in his early thirties), Boltzmann’s eyesight, poor from birth, was failing further. It became increasingly hard for him to do experiments, and as the years went by he was forced to rely increasingly, and then entirely, on the assistance of others in the lab.
Although Boltzmann had strong backing for the Graz position, an effort arose to bring Ernst Mach back from Prague, where he had now been for almost 10 years. Mach had, as it happened, married a young woman originally from Graz, while Boltzmann’s marriage to Henriette, set for July 17, gave him a sentimental reason to return to Graz as well.
Henriette found herself in a position to help her future husband, or at least deal in rumors and gossip. Because she lived in the household of Herr Kienzl, the mayor of Graz, she had connections with important people in the town and the university. She also knew Karl von Stremayr, a civil servant now with the Ministry for Education and Culture who had earlier worked with her father.
Not only that, she had a line of communication to the Mach camp, such as it was. The son of the mayor of Graz was one Wilhelm Kienzl, who during his lifetime became a well-known composer. He was a noted Wagnerian and his most famous work, an opera called Der Evangelimann (loosely, The Preacher) drew large audiences at its Berlin premiere in 1895. As a young man Kienzl was interested in both physics and music and had gone to study in Prague, where he took lectures from Mach. Mach’s candid assessment of Kienzl’s showing as a physics student solidified the young man’s determination to become a musician. In his autobiography, the composer presented a brief picture of Boltzmann at about this time, describing him as “a strong, heavy-browed man, very shortsighted and bespectacled on that account, with tightly curled brown hair and a full beard framing a broad, flushed face, always somewhat stooped in posture.”
In early June, only a few weeks before the wedding, Henriette was able to relay to Boltzmann a bit of news from Frau Kienzl, which was that she had spoken directly to Mach to let him know that Boltzmann was interested in the Graz position. Mach supposedly said that in that case, although he too would have liked the job, “if he had to make a proposal he would propose you as first choice.” Henriette also used her connection to Stremayr to intimate that her fiancé’s chest was inclined to be weak, and that the mountain air around Graz would be excellent for his health. She also reported that an official in Graz had told her that Mach is not likely to get the job because he is “industrious but no genius.”
The significance, if any, of these minor conspiracies is impossible to assess. In the very last week before their wedding day, Boltzmann’s agitation over the uncertainty boiled over. He began to worry that if, as planned, they were to go away to Switzerland on their honeymoon, and the decision were made in his absence, he might lose not only the Graz job but also some advantages of his position in Vienna. Abruptly, he proposed that after they get married, they should live in Vienna while things were worked out.
Boltzmann’s suggestion upset Henriette. “That all the beautiful hopes for our honeymoon should come to nothing!” she exclaimed. Rather than fall in with his new plan, she told him she would prefer to postpone the wedding altogether. Hastily and apologetically Boltzmann wrote back, saying that he had now had further words with someone in the ministry, who told him that a decision over Graz would not be made until the second half of August, so there was no point in postponing the wedding. “There is really nothing more I can do in Vienna,” he said.
So, five days later, on July 17, 1876, they were married in Graz as they had intended all along and went off to Switzerland for a honeymoon. Despite all his concerns, Boltzmann duly became professor of physics and director of the physics department in Graz. He was then 32 years old, and Henriette 22. Boltzmann had at first suggested to his fiancée that they should all live together, the newlyweds along with his mother and sister, but Henriette, as his wife, decided otherwise. The married couple found lodging for Boltzmann’s mother and sister elsewhere, while they at first lived in accommodations provided by the university.
Children soon began to arrive. Their first son and daughter, named Ludwig and Henriette after the parents, were born in 1878 and 1880. A second son, Arthur, came in 1881 and another daughter, Ida, in 1884. Their last child, Elsa, was born some years later, in 1891, after the Boltzmanns had left Graz. Despite her earlier strenuous attempts to study science, and despite Boltzmann’s insistence that his wife should be a comrade-in-arms and not a housewife, Henriette abandoned her academic plans, took cooking lessons from Frau Kienzl, and embarked on a life of seemingly typical domesticity. The Boltzmanns built themselves a house on farmland a few miles northeast of Graz, on the slopes of a mountain with splendid views over the surrounding country. Boltzmann doted on his children. He took them on walks in the countryside around Graz, teaching them the plants and flowers of the region. (A colleague at Graz, a professor of botany, said later how impressed he was with Boltzmann’s knowledge.) His devotion to his children took him on one occasion to an impractical extreme: he decided his daughters needed fresh milk, so he bought a cow at the local market and walked it home through the streets of Graz. But then he had to consult a zoology professor at the university to find out what one should feed a cow, and how one should arrange things so that it produced milk.
His years in Graz were the most productive of Boltzmann’s career. At just about the time he left Vienna, his earlier work in kinetic theory finally began to gain some broader attention—albeit critical attention—and as he defended his ideas, Boltzmann elaborated them further. But at the same time he continued to work in the laboratory and to cast his gaze at other issues in mathematical physics. In one notable achievement, he was able to repay the debt he owed to his first teacher, Josef Stefan, who had instilled in him an interest in electromagnetic phenomena in general and Maxwell’s theory in particular.
Stefan, in 1879, had established experimentally that electromagnetic radiation in thermal equilibrium had an intrinsic energy proportional to the fourth power of the temperature. In 1884, Boltzmann used Maxwell’s theory in combination with his sophisticated understanding of heat and energy to propound a theoretical explanation of this relation, and also to prove that the radiation would exert a pressure proportional to the cube of its temperature. This result, in which Boltzmann established a fundamental connection between radiation theory and thermodynamics, is now known as the Stefan-Boltzmann law.
Once he had his own institute in Graz, Boltzmann tried to reproduce the happy atmosphere he had enjoyed in his younger days in Erdbergstrasse, and to some extent he succeeded. As his name became known across Europe, a few students came to Graz specifically to study with him. Walther Nernst, who was to win the 1920 Nobel Prize in chemistry, visited in 1885 and was at first disappointed to find Boltzmann so busy with his introductory lectures in mathematics and physics that he had little time to spare. But then Boltzmann suggested an experimental investigation that might be worth doing, and once Nernst was embarked on this advanced project, he found Boltzmann happy to spend hours discussing its finer points. Nernst remembered a “well-organized institute, in which teachers and researchers worked together with students in an exemplary fashion.”
A similarly warm recollection came from Svante Arrhenius, who came to Graz from Sweden in 1887 and who was a Nobel Prize winner for chemistry in 1903. Like Nernst, he recalled Boltzmann being willing to discuss and debate science at great length with advanced students, but admitted that only a few enjoyed such close contacts. In the end, Boltzmann did not establish a school of his own with anything like the influence of Stefan’s Erdbergstrasse institute. In part, the students at Graz were not of such high quality or ambition as those in Vienna, but in part it seemed that Boltzmann would only engage directly those few promising young scholars who made the extra effort to seek him out.
Academic and civil distinctions came his way. He was made “government councilor,” an honorific of the Habsburg court, and some years later “court councilor.” He became a full member of the Viennese Academy of Sciences, and foreign academies honored him with membership. But at the same time Boltzmann would often feel isolated in Graz, away from the great academic centers of Europe. Besides Nernst and Arrhenius, he had little influence on young researchers destined for greatness. For several years after settling in Graz with his new wife, he hardly traveled at all, even to Vienna, and although he kept up a mostly personal correspondence with Toepler, he had only irregular contact with more notable researchers in Austria and Germany.
Even in the small world of Graz itself he had reclusive tendencies. His academic position as well as Henriette’s childhood in the Kienzl household gave the couple an automatic status in social circles, but they seemed to have little use for such things. One faculty member recalled that “with the physics genius Ludwig Boltzmann, who at that time already stood at the height of his glory, I like the rest of my colleagues had little personal contact, on account of the withdrawn way this shy eccentric lived.” Wilhelm Kienzl, the composer, offered a similar assessment. Depicting Boltzmann rather as Boltzmann depicted Loschmidt, Kienzl called him “the prototypical unworldly scholar, living wholly in the realm of his science and his groundbreaking research. . . . He commanded a broad range of general knowledge, which had no impact whatever on the manifestly childish naivete of his nature, as one often finds with those whose focussed minds move in higher spheres.”
For over a decade Boltzmann lived and worked in Graz, his scientific output steady, his name known increasingly throughout the scientific world, but maintaining little direct contact with his scientific colleagues elsewhere, and in person living with his family almost in solitude.