Fermat and Pascal
by Chetan Parikh
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In a great book, “Games, Gods and Gambling”, the author, F. N. David, writes about Fermat and Pascal.


“The flood of ideas generated by the Italian Renaissance did not abate after the death of Galileo, but it was apparent even in Galileo's lifetime that pre-eminence in both the arts and sciences was passing from Italy to France. The Renaissance in France was later than that of Italy and was probably greatly helped by the loot in the form of ideas, manuscripts and books which successive French armies carried out of Italy. The fabulous library of manuscripts of Petrarch is said to have been looted by the French in 1501, and this must have been only one of many instances. By the year 1600 the revolution in the arts and the development of the sciences had both reached mature proportions, although they did not produce any practitioner of the experimental method of the calibre of Galileo or Leonardo. The major contributions of the French scientists of the seventeenth century, when one surveys all the work done, might be said to be in the realms of pure and of applicable mathematics.


Galileo wrote as though the calculation of a probability was something which was obvious, and the suggestion from his work is that almost any mathematician could set out the method. Although this fragment was left unpublished by him, it does appear likely that the calculation of a probability was a common- place to the Italian mathematicians and probably therefore to some of those in France. Thus, while the foundation of the calculus of probabilities, the giant step forward made with the concept of the equally-likely faces of the die, does not belong to the French, the next development of the theory is undoubtedly theirs. And by far the greatest of the French mathematicians of the seventeenth century was Pierre de Fermat.


Fermat, born in 1601 at Beaumont-de-Lomagne near Montauban in Gascony-five years after Descartes and four years before Rembrandt-has been called by some the prince of amateurs and by others the greatest pure mathematician who has ever lived. There can be no doubt that, in an age when mathematical theories in general were being developed at a great rate, he was outstanding, and this quiet lawyer did more than any other Frenchman in helping the formulation of the theory of probability. Fermat was the son of Dominique Fermat, citizen and second consul of the town of Beaumont, and his wife Francoise de Cazelneuve. His life offers few noticeable incidents. His family were leather merchants and he spent his childhood at home. He studied law at the University of Toulouse and after passing his examinations was named Conseilleur de la Chambre des Requstes du Parlement of Toulouse in May, 1631, at the age of 30. Some days after he was given this position he married Louise du Long, daughter of a Counsellor in the same Parlement, and it is from this date that he assumed the prefix de. It is not known whether he was actually ennobled by a special decree or whether the post of Counsellor carried the implicit right to the prefix.


In the intervals when the law courts went into recess he studied both literature and mathematics. Parliamentary Counsellors of that age, like the judges of today, were obliged to hold themselves aloof from and to have very little contact with, their fellow citizens, and this must have helped to produce the necessary time for reflection. His biographers speak of his “singular erudition” in what would now be called the humanities. His knowledge of the chief European languages and of the literature of continental Europe was said to be both wide and accurate. He made emendations to Greek and Latin texts. He wrote verses in Latin, French and Spanish and he did research in mathematics. This quiet friendly man passed all his working life in the service of the State. He was promoted King’s Counsellor, still in the Parlement of Toulouse, in 1648, and he died at Castres in January 1665, when he was 64 years of age. His private life seems to have been as uneventful and successful as his public career. He had two daughters, both of whom became nuns, and three sons one of whom, Samuel, became known as a writer. Even after three hundred years the good temper, modesty, kindliness and intellectual brilliance of this great man shine through his letters which have now, fortunately, been collated, dated and printed in full.


Unlike the Italians of the sixteenth and seventeenth centuries and the French, Swiss and Italians of the eighteenth century, the public challenge to the problem does not seem to have played much part among Fermat and his contemporaries. It has been mentioned how Fiore, the pupil of Ferreo, gained a great reputation for himself and his teacher by challenging other mathematicians to solve problems concerning the roots of a cubic equation, and later John Bernoulli used this device to further his own reputation, but public disputation does not seem to enter just at this time. It would appear rather that the solution of a problem was followed by a letter to a friend telling him about it, and, possibly just to puzzle him a little, a step in the proof is held back, but this was done privately. Letters were also exchanged setting out the failure to solve a problem and asking for enlightenment. In all the vast correspondence of Fermat there appears only one suggestion that the correspondent might have been resentful of Fermat succeeding where he himself had failed.


This correspondent was Pascal. The development of mathematics in seventeenth-century France is interesting in that so very little was made public in comparison with what was achieved. If a scientist belonged to the closed circle then he corresponded with those others of the circle about anything and everything: but the comments and approbation of his equals seem to have been sufficient for him, and it was unusual for their letters to find their way into print. It is only because so many of these letters have survived that it is possible nowadays to give Fermat the credit which is certainly his right. On Fermat's part the lack of desire to publish may have been his modesty. He did not have the day-to-day contact which the group of mathematicians had in Paris and which must have supplemented their letters to each other. He seems to have been one of those rare persons-like Newton-who flourished in isolation and who was modest enough to believe that the sketch of a proof or even the statement of a theorem was enough for all the world to understand.


Fermat carried on an immense correspondence with scientists in Paris, but there are also records of letters going to the Low Countries, to Italy and sometimes to England. This century, the seventeenth, is noteworthy for the formal founding of a great number of scientific societies, and in this correspondence between the scientists of all nations we have the nuclei of these. Harcourt-Brown states that the Italian academies or societies date from the fifteenth century, and possibly the French borrowed this idea from them; or the societies may just have been the inevitable consequence of the liberation of thought following the slow climb of humanity from the Dark Ages. In France the academies were not so much social (as in Italy) as the regular meetings of scientists with comparable interests who exchanged ideas about the scientific problems of the day. At about the time of Fermat's birth one of the links between the academies of Italy and Paris was Nicolas-Claude Fabri de Peiresc (1580-1637). He is described as being inquisitive and very curious and was the ideal person to spread the gossip of new ideas and techniques. He had been a student at Padua and had contacts with the scholars of Firenze and other Italian centres of learning. During the years 1605-1620 he travelled extensively all over Europe, spreading his news as he went, and since one of his loves was astronomy he would have given news of Galileo. He also carried on a lively correspondence with the Abbe Mersenne. From the point of view of the probability calculus he is interesting only as a typical example of the way in which mathematical and other ideas were carried about Europe and because of his contact with the Mersenne Academy. Yet another possibility whereby the ideas about chance may have been propagated is Cardinal Francis Barberini who was liberal in his interpretation of Galileo's house-arrest and who was also a correspondent of the Abbe. So many of the threads from divers points in Italy are tied together on their arrival at the Academy of the Abbe Mersenne that the probabilities would be in favour of ideas about chance having come to this group from Italy unless they had arrived earlier and before its foundation. The group is a noteworthy one.


Marin Mersenne* was born in 1588 at the village of La Soultiere, near Oize (Sarthe), and educated at the Jesuit Seminary at La Fleche. He became a priest in the order of the Minorites and was stationed in Paris for much of his life. It is said that from his earliest years he was interested in music, in mathematics and in the natural sciences and he corresponded with nearly every scientist of note of his day. Harcourt-Brown writes:


There is hardly a figure of importance in the learned world who does not appear in the pages of his letters. From all parts of Europe news of the advancement of the sciences came to the convent “des peres Minimes, proche la Place Royale” and thence went the prized letters of the reverend father, written in their own peculiar, cramped and all but illegible hand, with the precious news of Descartes, Morin, Fermat, Torricelli or Galileo.


To Mersenne's house once a week came mathematicians and natural scientists, among them Gassendi, Desargues, Carcavi, Roberval, Descartes and the Pascals, father and son. The bond appears to have been Mersenne.


All who came into contact with him remarked on his universal learning, on the sweetness and charm of his speech, the gentleness of his temper, the naivete which won its way into all hearts.


Mersenne died in 1648 and the group changed its venue to the house of La Pailleur. After he died in 1651 they met in other places. These informal societies died with the formal beginnings of the learned societies, the Royal Society of London in 1660 and the Academic des Sciences of Paris in 1665. It would seem likely that Fermat was au courant with all that was being discussed in the scientific world both through his own correspondence and through the information which he would gain through the Mersenne Academy. It is almost certain that if the fragment of Galileo on dicing had been thought sufficiently new to discuss, the informal academies would have done so within a few years of its having been written.


The name of Blaise Pascal is always linked with that of Fermat as one of the “joint discoverers” of the probability calculus. Because his mathematical work came in bursts before he retired at an early age to meditate on “the greatness and the misery of man” and was negatively correlated with these meditations, it is worthwhile to consider briefly the outline of his life. Etienne Pascal was a judge at Clermont-Ferrand in the Auvergne and is described as a very learned man and an able mathematician. In 1631, having, so his biographers say, “an extraordinary tenderness for his child, his only son,” he gave up his post as judge and moved to Paris in order to supervise his son's education. He became a member of Mersenne's Academy, although little is recorded of his part in their discussions. Many sources agree, however, that he did take an active part. Carcavi, who, like Fermat, was a Counsellor in the Parliament of Toulouse, was a correspondent of Mersenne before he moved to Paris in 1642 and became a member of the Academy. He was responsible for introducing Fermat into the correspondence circle of Mersenne in 1636, and it was at his suggestion that Etienne Pascal and Roberval wrote to Fermat (April 26th, 1636) concerning the weight of the earth. Roberval and Pascal attacked Fermat's theory and there was an exchange of letters about it. The relations of Fermat with the Academy remained, however, excellent, and when in 1637 Descartes attacked Fermat's method of maximum and minimum tangents Roberval and Pascal were Fermat's defenders, supporting him with a polemic which is well known. The elder Pascal introduced Blaise into the academy when he was fourteen years old, that is, about the year 1637 when the controversy was at its height.


Blaise Pascal (1623-1662), born when Descartes was 27 years of age, appears to have been of poor physique and of precocious mental ability. His father, it is stated, from an early age concentrated on developing his reasoning powers rather than his memory. Blaise was undoubtedly quick-witted. His sister, married to a physicist, M. Perier, wrote a life of her brother which is not without a certain imaginative interest. She says her brother had an extraordinary wit at a very young age which he showed by repartee quick and to the point. She also relates that Etienne Pascal was afraid of his son overtaxing his strength, and while he instructed him in what is described as the usual education of the time (presumably classical languages, commentaries on Plato and Aristotle and so on), he hid all books on mathematics from his son. One day he found him drawing diagrams in charcoal and saw that he had rediscovered for himself Euclid's propositions.* Now although Madame Gilberte Perier was imaginative with regard to her brother and while much that she writes may be discounted for this reason, there is no doubt that Blaise could invent for himself without prompting of book or person. For when he was sixteen (c. 1639) he wrote his famous Essai pour les Coniques which, although no longer surviving as a whole, certainly did exist since Leibnitz reports having seen it. Blaise’s work caused a certain amount of controversy; some mathematicians received it with acclamation, while others, among them Descartes, refused to believe that it was entirely his own work. While granting more to Pascal than Descartes was willing to do, it may be that he was right to be a little sceptical. The method of projection used had been put forward previously by Desargues, and while the creative thought for the essay probably came from Blaise we may wonder how far Etienne acted as an improver or refiner of his son's work. The maturity of the essay, which caused Descartes to be suspicious of its authorship, was possibly due to Etienne. But even so, allowing for his father’s improvements and Desargue’s trail- breaking in the form of method, enough is left to demonstrate the astonishing mathematical powers of this boy of sixteen.


Two years later Blaise invented a calculating machine, but his health, already delicate, deteriorated and he had to give up working for four years. This is possibly the worst thing which could have happened to this delicate introspective boy who wanted to know the reason of everything, and who, when adequate reasons were not given him, looked for better ones for himself: Four or five years after his breakdown, in 1646, we hear of his conversion to Jansenism in company with the rest of his family. This cult of Jansenism affected many scientists of the seventeenth century and was the ostensible* reason for the failure of Pascal to fulfil his mathematical promise, so I will consider the history of the cult briefly and its destiny. Without attempting to disentangle the theological doctrines involved, and thereby being quite unjust, it would appear that the cult of Jansenism arose from a violent dislike (and possibly envy) of various Catholic bishops and heads of seminaries for the power of the Society of Jesus. This Society, founded in 1534, was growing all-powerful, and many Catholics of importance felt dislike of the rate at which its influence was increasing. Within the framework of Catholic dogma, therefore, they set themselves to attack the Jesuits on the subjects of freewill and of the grace of God. The Jansenists took what has been described as a standpoint akin to Calvinism. Cornelius Jansen, created Bishop of Ypres in 1636, was the leader of this sect and it was the tract written by Jansen on Reformation de I’Homme Interieur which is held to have been the cause of Pascal's conversion. Blaise appears to have become interested in the sect when his father was nursed by the Jansenists during an illness at Rouen, and he finally retired to die at Port-Royal. The celebrated Cistercian Abbey of Port-Royal, built in the valley of the Yvette 30 miles west of Paris in the village of Les Hameaux, was founded in 1204 by the wife of a French nobleman when he was absent from France on the Fourth Crusade. The Pope gave this abbey the privilege of affording a retreat to lay persons who wanted to withdraw from the world for a time but who did not want to bind themselves with permanent vows. A second abbey of the same foundation was instituted in Paris in 1626, and it was to this Abbey that Pascal retired for meditation, the last time for good.


Jansen died in 1638 and his final apologia was printed by his friends after his death (1640). The real struggle between the Jesuits and the Jansenists now began, with the Jesuits trying to persuade successive popes to declare the Jansenists heretics and to excommunicate them, and with the Jansenists preaching and pamphleteering against the Jesuits. The Sorbonne as a whole seems to have been moderately inclined to Jansenism. Bishop Jansen in his Discours held that scientific curiosity was only another form of sexual indulgence. “On reading this page,” writes Sainte-Beuve, an eminent biographer of Pascal, “a curtain was drawn in Pascal's soul. Physics, geometry appeared to him for the first time in a new light.” This first conversion of Blaise in 1646, when he was 23, does not seem to have lasted very long, for in 1648 he took up mathematics and physics again. Perier, at Pascal's suggestion, carried out the famous experiments with a barometer at the Puy de Dome. The conclusions which Blaise drew from these experiments he wrote in La Pesanteur de l’Air and thus involved himself in further dispute with Descartes. Descartes had discussed the possibility of these experiments in letters to Mersenne, and Pascal would have been privy to these since he was a more or less regular attendant at the academy. The scepticism of Descartes with reference to Blaise's essay on conics has already been noticed. He was put out by these experiments of Perier’s, and this was probably accentuated by the fact that Descartes (like Mersenne) was a protege of the Jesuits and a great lover of the Society of Jesus. In the long run he does not seem to have borne Blaise any malice, but the incident seems to show that Blaise is not entirely the originator which posterity is inclined to believe him. After the controversy Blaise wrote to his sister Jacqueline that as a result of his terrible struggle and the indulgence of his scientific curiosity he had become paralysed and was able to walk only with crutches. This paralysis did not however last long.


In 1651 Etienne died, leaving Blaise a moderate fortune: one of his sisters was married and the other had become a nun. The leading strings were at last removed and in 1653 he is described as a man of the world, leading a dissolute life. Whether he was exceptionally wild or whether he merely led the life of any young man of that time it is not possible to say, but whatever the truth of the matter his delicate health (probably) brought him up short before many months had elapsed. In 1651 he had the famous correspondence with Fermat on the problem of Points. Letters passed between the two scientists during the four months July to October, and the correspondence was definitely closed by Pascal before his second conversion on November 23rd, 1654. It is said that the horses of a four-in-hand ran away with him: he took this as a sign from God that he must give up the life he was leading and do no more mathematics, and he retired to Port-Royal. The story of the rest of his life is concerned more with Jansenism than with mathematics. At Port-Royal he could not still his restless enquiring mind and before long he took up the cudgels for the Jansenists against Pope Innocent X. The Sorbonne had now decided that it was politic to regard the Jansenists as heretics. Pascal published in January 1656 the first two of his famous Provincial letters, of which Voltaire wrote: “A book of genius is seen in Les Lettres Provinciales. All types of eloquence are to be found in them: there is not a single word which after 100 years should be changed.” Voltaire was, however, anti-Jesuit himself. At the time (1656) the letters made a great impression but could not save the Professor of Theology (Antoine d’Arnauld) from being expelled from the Sorbonne.


The Jansenists were persecuted, their leaders were forced to go into hiding, and the nuns of Port-Royal were subjected to imprisonment. Pascal continued to live in extremely ascetic circumstances, spending his time reading the Scriptures and in writing down the thoughts which these spiritual exercises evoked. These thoughts were published after his death and form the famous Pensees. It is related that in 1658 he had toothache which kept him awake, and that to distract himself he thought about the cycloid, the curve traced out by a fixed point on the circumference of a wheel rolling at a uniform speed on a horizontal plane. As he thought, the pain disappeared. He took this to be a sign that the Almighty didn't mind him thinking about the cycloid, so he thought about it for eight days and wrote his results to Carcavi. The fact that he wrote under a pseudonym is possibly a sign that he did not want his world to know that he had fallen so far from grace as to indulge in the sin of mathematical research. He died in 1662, at the age of 39, from convulsions, the post- mortem showing that he had a serious lesion in the brain. He was buried in the church of St. Etienne-du-Mort.”