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Do you not feel and hear it? Do I alone hear this melody so wondrously and gently sounding … Richard Wagner. Tristan und Isolde (Act III, Scene iii)

In 1809 Wilhelm von Humboldt became the education minister for the north German state of Prussia. In a letter to Goethe in 1816 he wrote, ‘I have busied myself here with science a great deal, but I have deeply felt the power antiquity has always wielded over me. The new disgusts me …’ Humboldt favoured a movement away from science as a means to an end, and a return to a more classical tradition of the pursuit of knowledge for its own sake. Previous education schemes had been geared to providing civil servants for the greater glory of Prussia. From now on, more emphasis was to be placed on education serving the needs of the individual rather than the state.

In his role as a thinker and civil servant, Humboldt enacted a revolution with far-reaching effects. New schools, called Gymnasiums, were created across Prussia and the neighbouring state of Hanover. Eventually the teachers in these schools were not to be members of the clergy, as in the old education system, but graduates of the new universities and polytechnics that were built during this period.

The jewel in the crown was Berlin University, founded in 1810 during the French occupation. Humboldt called it the ‘mother of all modern universities’. Housed in what had once been the palace of Prince Heinrich of Prussia on the grand boulevard Unter den Linden, the university would for the first time promote research alongside teaching. ‘University teaching not only enables an understanding of the unity of science but also its furtherance,’ Humboldt declared. Despite his passion for the Ancient World, under his guidance the university pioneered the introduction of new disciplines to sit beside the classical faculties of Law, Medicine, Philosophy and Theology.

For the first time, the study of mathematics was to form a major part of the curriculum in the new Gymnasiums and universities. Students were encouraged to study mathematics for its own sake and not simply as a servant of the other sciences. This contrasted starkly with Napoleon’s educational reforms, which saw mathematics harnessed to further French military aims. Carl Jacobi, one of the professors in Berlin, wrote to Legendre in Paris in 1830 about the French mathematician Joseph Fourier, who had reproached the German school of thought for ignoring more practical problems:

It is true that Fourier was of the opinion that the principal object of mathematics is public use and the explanation of natural phenomena; but a philosopher like him ought to have known that the sole object of the science is the honour of the human spirit, and that on this view a problem in the theory of numbers is worth as much as a problem of the system of the world.

For Napoleon, it was education that would finally destroy the arcane rules of the ancien régime. His recognition that education was the backbone for building his new France had led to the establishment in Paris of some of the institutes which are still famous today. Not only were the colleges meritocratic, allowing students from all backgrounds to attend, but also the educational philosophy put a greater emphasis on education and science serving society. One of the French Revolutionary regional officers wrote to a professor of mathematics in 1794, commending him on teaching a course in ‘Republican arithmetic’: ‘Citizen. The Revolution not only improves our morals and paves the way for our happiness and that of future generations, it even unlooses the shackles that hold back scientific progress.’

Humboldt’s approach to mathematics was very different from this utilitarian philosophy that prevailed across the border. The liberating effect of Germany’s educational revolution was to have a great impact on mathematicians’ understanding of many aspects of their field. It would allow them to establish a new, more abstract language of mathematics. In particular, it would revolutionise the study of prime numbers.

One town that benefited from Humboldt’s initiatives was Lüneberg, in Hanover. Lüneberg, once a thriving commercial centre, was now a town in decline. Its narrow streets paved with cobblestones were no longer buzzing with the business it had seen in previous centuries. But in 1829 a new building was erected amidst the tall towers of the three Gothic churches in Lüneberg: the Gymnasium Johanneum.

By the early 1840s the new school was flourishing. Its director, Schmalfuss, was a keen proponent of the neo-humanist ideals initiated by Humboldt. His library reflected his enlightened views: it featured not only the classics and the works of modern German writers, but also volumes from farther afield. In particular, Schmalfuss managed to get his hands on books coming out of Paris, the powerhouse of European intellectual activity during the first half of the century.

Schmalfuss had just accepted a new boy at the Gymnasium Johanneum, Bernhard Riemann. Riemann was very shy and found it difficult to make friends. He had been attending the Gymnasium in the town of Hanover, where he had been boarding with his grandmother, but when she died, in 1842, he was forced to move to Lüneberg where he could board with one of the teachers. Joining the school after all his contemporaries had established friendships did not make life easy for Riemann. He was desperately homesick and was teased by the other children. He would rather walk the long distance back to his father’s house in Quickborn than play with his contemporaries.

Riemann’s father, the pastor in Quickborn, had high expectations for his son. Although Bernhard was unhappy at school, he worked hard and conscientiously, determined not to disappoint his father. But he had to battle with an almost disabling streak of perfectionism. His teachers would often get frustrated at Riemann’s inability to submit his work. Unless it was perfect, the boy could not bear to suffer the indignity of anything less than full marks. His teachers began to doubt whether Riemann would ever be able to pass his final examinations.

It was Schmalfuss who saw a way to bring the young boy on and exploit his obsession with perfection. Early on, Schmalfuss had spotted Riemann’s special mathematical skills and was keen to stimulate the student’s abilities. He allowed Riemann the freedom of his library, with its fine collection of books on mathematics, where the boy could escape the social pressures of his classmates. The library opened up a whole new world for Riemann, a place where he felt at home and in control. Suddenly here he was in a perfect, idealised mathematical world where proof prevented any collapse of this new world around him, and numbers became his friends.

Humboldt’s drive from teaching science as a practical tool to the more aesthetic notion of knowledge for its own sake had filtered down to Schmalfuss’s classroom. The teacher steered Riemann’s reading away from mathematical texts full of formulas and rules that were aimed at feeding the demands of a growing industrial world, and guided him towards the classics of Euclid, Archimedes and Apollonius. With their geometry, the ancient Greeks sought to understand the abstract structure of points and lines, and they were not hung up on the particular formulas behind the geometry. When Schmalfuss did give Riemann a more modern text, Descartes’s treatise on analytical geometry – a subject rife with equations and formulas – the teacher could see that the mechanical method developed in the book did not appeal to Riemann’s growing taste for conceptual mathematics. As Schmalfuss later recalled in a letter to a friend, ‘already at that time he was a mathematician next to whose wealth a teacher felt poor’.

One of the books that was sitting on Schmalfuss’s shelf was a contemporary volume the teacher had acquired from France. Published in 1808, Théorie des Nombres by Adrien-Marie Legendre was the first text to record the observation that there seemed to be a strange connection between the function that counted the number of prime numbers and the logarithm function. This connection, discovered by Gauss and Legendre, was only based on experimental evidence. It was far from clear whether, as one counted higher, the number of primes would always be approximated by Gauss’s or Legendre’s function.

Despite the volume’s 859 large quarto pages, Riemann gobbled it up, returning the book to his teacher just six days later with the precocious declaration, ‘This is a wonderful book; I know it by heart.’ His teacher could not believe it, but when he examined Riemann on its contents during his final examinations two years later, the student answered perfectly. That marked the beginning of the career of one of the giants of modern mathematics. Thanks to Legendre, a seed was sown in the young Riemann’s mind that in later life was to blossom in spectacular fashion.

His final examinations over, Riemann was eager to join one of the vigorous new universities that were driving the educational revolution in Germany. His father, though, had other ideas. Riemann’s family was poor, and his father hoped that Bernhard would join him in the Church. The life of a clergyman would provide him with a regular income with which he could support his sisters. The only university in the Kingdom of Hanover to offer theology was not one of these new establishments but the University of Göttingen, founded over a century before, in 1734. So in 1846, to comply with his father’s wishes, Riemann made his way to the dank town of Göttingen.

Göttingen sits quietly in the gentle hills of Lower Saxony. At its heart lies a small medieval town enclosed by ancient ramparts. This is the Göttingen that Riemann would have known, and it still retains much of its original character. The streets wind narrowly between half-timbered houses topped with red-tiled roofs. The Brothers Grimm wrote many of their fairy tales in Göttingen, and one can imagine Hansel and Gretel running through its streets. In the centre stands the medieval town hall, whose walls display the motto ‘Away from Göttingen there is no life.’ For those at the university that was certainly the feeling. The academic life was one of self-sufficiency. Although theology had predominated in the early years of the university, the winds of academic change sweeping across Germany had stimulated Göttingen’s scientific curriculum. By the time Gauss was appointed as professor of astronomy and director of the observatory in 1807, it was science rather than theology for which Göttingen was becoming famous.

The fire for mathematics that the teacher Schmalfuss had ignited in the young Riemann was still burning strong. His father’s wish that he study theology had brought him to Göttingen, but it was the influence of the great Gauss and Göttingen’s scientific tradition that left its mark during that first year. It was only a matter of time before Greek and Latin lectures gave way to the temptation of courses in physics and mathematics. With trepidation, Riemann wrote to his father suggesting that he would like to switch from theology to mathematics. His father’s approval meant everything to Riemann. With a sense of relief he received his father’s blessing, and immediately immersed himself in the scientific life of the university.

To a young man of such talent, Göttingen soon began to feel small. Within a year Riemann had exhausted the resources available to him. Gauss, by now an old man, had become quite withdrawn from the intellectual life of the university – since 1828 he had spent only one night away from the observatory, where he lived. At the university he only lectured on astronomy, and in particular on the method that had made him famous when he’d rediscovered the ‘lost’ planet Ceres many years before. Riemann had to look elsewhere for the stimulus he needed to take the next step in his development. He could see Berlin was where the buzz of intellectual activity was the loudest.

The University of Berlin had been greatly influenced by the successful French research institutes, such as the École Polytechnique, that had been founded by Napoleon. It had, after all, been founded during the French occupation. One of the key mathematical ambassadors was a brilliant mathematician by the name of Peter Gustav Lejeune-Dirichlet. Although he was born in Germany in 1805, Dirichlet’s family was of French origin. A return to his roots took him to Paris in 1822, where he spent five years soaking up the intellectual activity that was bubbling out of the academies. Wilhelm von Humboldt’s brother Alexander, an amateur scientist, met Dirichlet on his travels and was so impressed that he secured him a position back in Germany. Dirichlet was something of a rebel. Perhaps the atmosphere on the streets in Paris had given him a taste for challenging authority. In Berlin, he was quite happy to ignore some of the antiquated traditions demanded by the rather stuffy university authorities, and often flouted their demands to demonstrate his command of the Latin language.

Göttingen and Berlin offered emerging scientists such as Riemann contrasting academic climates. Göttingen revelled in its independence and isolation. Few seminars were presented by anyone from beyond the city walls. It was self-sufficient and generated great science from the fuel burning within. Berlin, on the other hand, thrived on the stimulation coming from outside Germany. The ideas feeding through from France mixed with the forward-looking German approach to natural philosophy to create a heady new cocktail.

The different climates of Göttingen and Berlin suited different mathematicians. Some would never have succeeded without contact with new ideas from external sources. The success of other mathematicians can be traced back to an isolation which forced them to find an inner strength and new languages and modes of thought. Riemann would turn out to be someone whose breakthroughs came from contact with the wealth of new ideas that were in the air, and he could see that Berlin was the place to be.

Riemann made his move in 1847 and remained in Berlin for two years. While there, he was able to get his hands on papers by Gauss which he had not been able to prise from the reticent master in Göttingen. He attended lectures by Dirichlet, who was later to play a part in Riemann’s dramatic discoveries about prime numbers. By all accounts, Dirichlet was an inspiring lecturer. One mathematician who attended his lectures put it thus:

Dirichlet cannot be surpassed for richness of material and clear insight … he sits at the high desk facing us, puts his spectacles up on his forehead, leans his head on both hands, and … inside his hands he sees an imaginary calculation and reads it out to us – that we understand it as well as if we too saw it. I like that kind of lecturing.

Riemann made friends with several young researchers in Dirichlet’s seminars who were equally fired up by their passion for mathematics.

Other forces were also bubbling away in Berlin. The revolution of 1848 that swept away the French monarchy spread from the streets of Paris throughout much of Europe. It found its way to the streets of Berlin while Riemann was studying there. According to accounts of his contemporaries, it had a profound impact on him. On one of the few occasions in his life on which he joined with those around him on anything other than an intellectual level, he enlisted in the student corps defending the king in his Berlin palace. It is reputed that he did a continuous sixteen-hour stint on the barricades.

Riemann’s response to the mathematical revolution spreading from the Paris academies was not that of a reactionary. Berlin was importing not only political propaganda from Paris, but also many of the prestigious journals and publications coming out of the academies. Riemann received the latest volumes of the influential French journal Comptes Rendus and holed himself up in his room to pore over papers by the mathematical revolutionary Augustin-Louis Cauchy.

Cauchy was a child of the Revolution, born a few weeks after the fall of the Bastille in 1789. Undernourished by the little food available during those years, the feeble young Cauchy preferred to exercise his mind rather than his body. In time-honoured fashion, the mathematical world provided a refuge for him. A mathematical friend of Cauchy’s father, Lagrange, recognised the young boy’s precocious talent and commented to a contemporary, ‘You see that little young man? Well! He will supplant all of us in so far as we are mathematicians.’ But he had interesting advice for Cauchy’s father. ‘Don’t let him touch a mathematical book till he is seventeen.’ Instead, he suggested stimulating the boy’s literary skills so that when eventually he returned to mathematics he would be able to write with his own mathematical voice and not one he had picked up from the books of the day.

It proved to be sound advice. Cauchy developed a new voice that was irrepressible once the floodgates protecting Cauchy from the outside world had been reopened. Cauchy’s output grew to be so immense that the journal Comptes Rendus had to impose a page limit on articles it printed that is strictly adhered to even today. Cauchy’s new mathematical language was too much for some of his contemporaries. The Norwegian mathematician Niels Henrik Abel wrote in 1826, ‘Cauchy is mad … what he does is excellent but very muddled. At first I understood practically none of it; now I see some of it more clearly.’ Abel goes on to observe that of all the mathematicians in Paris, Cauchy was the only one doing ‘pure mathematics’ whilst others ‘busy themselves exclusively with magnetism and other physical subjects … he is the only one who knows how mathematics should be done’.

Cauchy was to land himself in trouble with the authorities in Paris for steering students away from practical applications of mathematics. The director of the École Polytechnique, where Cauchy was lecturing, wrote to him criticising him for his obsession with abstract mathematics: ‘It is the opinion of many persons that instruction in pure mathematics is being carried too far at the École and that such an uncalled for extravagance is prejudicial to the other branches.’ So it was perhaps no wonder that Cauchy’s work would be appreciated by the young Riemann.

So exciting were these new ideas that Riemann almost became a recluse. His contemporaries were to see nothing of him while he waded through Cauchy’s output. Several weeks later Riemann resurfaced, declaring that ‘this is a new mathematics’. What had captured Cauchy and Riemann’s imagination was the emerging power of imaginary numbers.