Fermat’s Last Theorem. Simon Singh

were elevated to fill the places of those who died. Even Fermat suffered a serious bout of plague in 1652, and was so ill that his friend Bernard Medon announced his death to several colleagues. Soon after he corrected himself in a report to the Dutchman Nicholas Heinsius:

      I informed you earlier of the death of Fermat. He is still alive, and we no longer fear for his health, even though we had counted him among the dead a short time ago. The plague no longer rages among us.

      

      In addition to the health risks of seventeenth-century France, Fermat had to survive the political dangers. His appointment to the Parliament of Toulouse came just three years after Cardinal Richelieu was promoted to first minister of France. This was an era of plotting and intrigue, and everyone involved in the running of the state, even at local government level, had to take care not to become embroiled in the machinations of the Cardinal. Fermat adopted the strategy of performing duties efficiently without drawing attention to himself. He had no great political ambition, and did his best to avoid the rough and tumble of parliament. Instead he devoted all his spare energy to mathematics and, when not sentencing priests to be burnt at the stake, Fermat dedicated himself to his hobby. Fermat was a true amateur academic, a man whom E.T. Bell called the ‘Prince of Amateurs’. But so great were his talents that when Julian Coolidge wrote Mathematics of Great Amateurs, he excluded Fermat on the grounds that he was ‘so really great that he should count as a professional’.

      At the start of the seventeenth century, mathematics was still recovering from the Dark Ages and was not a highly regarded subject. Similarly mathematicians were not treated with great respect and most of them had to fund their own studies. For example, Galileo was unable to study mathematics at the University of Pisa and was forced to seek private tuition. Indeed, the only institute in Europe to actively encourage mathematicians was Oxford University which had established the Savilian Chair of Geometry in 1619. It is true to say that most seventeenth-century mathematicians were amateurs, but Fermat was an extreme case. Living far from Paris he was isolated from the small community of mathematicians that did exist, which included such figures as Pascal, Gassendi, Roberval, Beaugrand and most notably Father Marin Mersenne.

      Father Mersenne made only minor contributions to number theory and yet he played a role in seventeenth-century mathematics which was arguably more important than any of his more esteemed colleagues. After joining the order of Minims in 1611, Mersenne studied mathematics and then taught the subject to other monks and to nuns at the Minim convent at Nevers. Eight years later he moved to Paris to join the Minims de l’Annociade, close to the Place Royale, a natural gathering place for intellectuals. Inevitably Mersenne met the other mathematicians of Paris, but he was saddened by their reluctance to talk to him or to each other.

      The secretive nature of the Parisian mathematicians was a tradition which had been passed down from the cossists of the sixteenth century. The cossists were experts in calculations of all kinds and were employed by merchants and businessmen to solve complex accounting problems. Their name derives from the Italian word cosa, meaning ‘thing’, because they used symbols to represent an unknown quantity, similar to the way mathematicians use x today. All professional problem-solvers of this era invented their own clever methods for performing calculations and would do their utmost to keep these methods secret in order to maintain their reputation as the only person capable of solving a particular problem. On one exceptional occasion Niccolò Tartaglia, who had found a method for quickly solving cubic equations, revealed his discovery to Girolamo Cardano and swore him to absolute secrecy. Ten years later Cardano broke his promise and published Tartaglia’s method in his Ars Magna, an act which Tartaglia would never forgive. He broke off all relations with Cardano and a bitter public dispute ensued, which only served to further encourage other mathematicians to guard their secrets. The secretive nature of mathematicians continued right up until the end of the nineteenth century, and as we shall see later there are even examples of secret geniuses working in the twentieth century.

      When Father Mersenne arrived in Paris he was determined to fight against the ethos of secrecy and tried to encourage mathematicians to exchange their ideas and build upon each other’s work. The monk arranged regular meetings and his group later formed the core of the French Academy. When anyone refused to attend, Mersenne would pass on to the group whatever he could by revealing letters and papers – even if they had been sent to him in confidence. It was not ethical behaviour for a man of the cloth, but he justified it on the grounds that the exchange of information would benefit mathematics and mankind. These acts of indiscretion naturally caused bitter arguments between the well-meaning monk and the taciturn prima donnas, and eventually destroyed Mersenne’s relationship with Descartes which had lasted since the two men had studied together at the Jesuit College of La Flèche. Mersenne had revealed philosophical writings by Descartes which were liable to offend the Church, but to his credit he did defend Descartes against theological attacks, as in fact he had done earlier in the case of Galileo. In an era dominated by religion and magic Mersenne stood up for rational thought.

      Mersenne travelled throughout France and further afield, spreading news of the latest discoveries. In his travels he would make a point of meeting up with Pierre de Fermat and, indeed, seems to have been Fermat’s only regular contact with other mathematicians. Mersenne’s influence on this Prince of Amateurs must have been second only to the Arithmetica, a mathematical treatise handed down from the ancient Greeks which was Fermat’s constant companion. Even when he was unable to travel Mersenne would maintain his relationship with Fermat and others by writing prolifically. After Mersenne’s death his room was found stacked with letters written by seventy-eight different correspondents.

      Despite the encouragement of Father Mersenne, Fermat steadfastly refused to reveal his proofs. Publication and recognition meant nothing to him and he was satisfied with the simple pleasure of being able to create new theorems undisturbed. However, the shy and retiring genius did have a mischievous streak, which, when combined with his secrecy, meant that when he did sometimes communicate with other mathematicians it was only to tease them. He would write letters stating his most recent theorem without providing the accompanying proof. Then he would challenge his contemporaries to find the proof. The fact that he would never reveal his own proofs caused a great deal of frustration. Rene Descartes called Fermat a ‘braggart’ and the Englishman John Wallis referred to him as ‘That damned Frenchman’. Unfortunately for the English, Fermat took particular pleasure in toying with his cousins across the Channel.

      As well as having the satisfaction of annoying his colleagues, Fermat’s habit of stating a problem but hiding its solution did have more practical motivations. First, it meant that he did not have to waste time fully fleshing out his methods; instead he could rapidly proceed to his next conquest. Furthermore, he did not have to suffer jealous nit-picking. Once published, proofs would be examined and argued over by everyone and anyone who knew anything about the subject. When Blaise Pascal pressed him to publish some of his work, the recluse replied: ‘Whatever of my work is judged worthy of publication, I do not want my name to appear there.’ Fermat was the secretive genius who sacrificed fame in order not to be distracted by petty questions from his critics.

      This exchange of letters with Pascal, the only occasion when Fermat discussed ideas with anyone but Mersenne, concerned the creation of an entirely new branch of mathematics – probability theory. The mathematical hermit was introduced to the subject by Pascal, and so, despite his desire for isolation, he felt obliged to maintain a dialogue. Together Fermat and Pascal would discover the first proofs and cast-iron certainties in probability theory, a subject which is inherently uncertain. Pascal’s interest in the subject had been sparked by a professional Parisian gambler, Antoine Gombaud, the Chevalier de Méré, who had posed a problem which concerned a game of chance called points. The game involves winning points on the roll of a dice, and whichever player is the first to earn a certain number of points is the winner and takes the prize money.

      Gombaud had been involved in a game of points with a fellow-gambler when they were forced to abandon the game half-way through, owing to a pressing engagement. The problem then arose as to what to do with the prize money. The simple solution would have been to have given all the money to the competitor with the most points, but Gombaud asked Pascal if there was a fairer way to divide the money. Pascal was asked