# Pierre de Fermat

Pierre de Fermat
Pierre de Fermat on an engraving by François de Poilly the Elder

Pierre de Fermat [ pjɛːʀ dəfɛʀˈma ] (* in the second half of 1607 in Beaumont-de-Lomagne , today in the Tarn-et-Garonne department ; †  January 12, 1665 in Castres ) was a French mathematician and lawyer .

## Life

### Year of birth and parents

Rear of Fermat's birthplace in Beaumont-de-Lomagne, 16th century; in the back right the tower belonging to the house with a viewing platform, the second tallest building in Beaumont

For a long time, the date of birth was August 20, 1601. However, recent research has shown that Fermat was born in Beaumont-de-Lomagne , a bastide 55 km northwest of Toulouse, in late 1607 or early 1608 . "Piere (Pierre) Fermat", baptized on August 20, 1601 in Beaumont-de-Lomagne, was an early deceased half-brother of the same name from the first marriage of his father Dominique Fermat to Françoise Cazeneuve, who died in 1603. His father married Claire de Long for the second time in 1604, and (probably between October 30 and December 6) in 1607 the future mathematician Pierre Fermat was born. Fermat's father, Dominique Fermat, was a successful wholesaler in rural products (wheat, wine and cattle, also hides and skins, but no leather) who had achieved considerable wealth and a high reputation through great skill. He was elected three times by the citizens of Beaumont for one year each as one of the four consuls who administered the city (there was no mayor). His second marriage to the noble Huguenot Claire de Long from the Huguenot stronghold of Montauban reflected the social status he has now achieved.

### Education

Formerly the Collège de Navarre in Montauban

There are no reliable findings about Fermat's schooling, but plausible assumptions. He probably spent his first years of elementary school in one of the three elementary schools for boys in Beaumont. It is hardly possible that he attended the Franciscan College in his hometown, as you can read again and again, because this collège was only opened in 1683. The Franciscans , who had had a small convent in Beaumont since 1516, were also forbidden to teach Greek. Another variant, represented by Michael S. Mahoney , is that Fermat received his primary and secondary education in the Grandselve monastery (ten kilometers east of Beaumont) operated by the Cordeliers (Franciscans) . But Grandselve was a Cistercian monastery and although it had founded a collège, the collège Saint Bernard, in 1281 , only the monks of the order were taught theology in this.

Fermat's mother Claire de Long died in 1615, probably in childbed after the birth of her daughter Jeanne, who apparently did not survive the death of her mother either. Fermat therefore became a half-orphan when he was less than eight years old.

Fermat probably received his much-praised classical education from 1617 to 1623 in the reformed collège de Navarre in nearby Montauban. His maternal grandmother, Bourguine de l'Hospital, lived there. This collège was also attended by Samuel de Long, his mother's brother, during the last years of the wars of religion . It also took in Catholic students.

### Studied in Orléans

Salle des Thèses of the old Orléans University, where the students took their exams

Fermat studied civil law from 1623 to 1626 at the University of Orléans and graduated in July 1626 with the baccalaureus iuris civilis . His lifelong friendship with Pierre de Carcavi (also Carcavy) (1606? –1684), who later became his colleague at the parlement de Toulouse for a few years (1632–1636) , also dates from this period of study . The University of Orléans was an old school of law with university rank, famous throughout Europe, in which, in the first three decades of the 17th century, the mos gallicus (the humanisme juridique ) was taught by three notable professors: Raoul Fornier (1562 –1627), Jérome L'Huillier († 1633) and Jehan Matthieu Le Grand († 1625). Students who were not yet of legal age (Fermat was almost 16 at the start of their studies) were accepted en tutelle in the house of one of these scholars and also taught there. It is not known who received Fermat. But that he received the finishing touches to his widely admired classical education there is more than just speculation.

### Lawyer in Bordeaux

Handwritten note in the margin by Fermat in the Konika des Apollonios von Perge

In September 1626 Dominique Fermat designated his older son Pierre as a universal heir in his will, in which he stipulated a settlement for his younger son Clément (who presumably died of the plague in 1631/32 ) and a dowry for his daughters Louise and Marie. In the autumn of the same year, Pierre Fermat, presumably on the advice of the mathematician Jean Beaugrand (1584-1640), settled as a lawyer (avocat) at the parlement de Bordeaux , where he stayed until the end of 1630. In June 1626, Louis XIII. spent a week in Orléans with his court, in which Jean Beaugrand was also staying, before taking the entire entourage by boat to Nantes on the Loire. Beaugrand had good relations with a circle of mathematically interested men in Bordeaux (the names Etienne d'Espagnet (d'Espaignet), François Philon and Pierre Prades are known from Fermat's correspondence ) and should recommend Fermat to one of them, Etienne d'Espagnet , have given. D'Espagnet, a young conseiller at the parlement de Bordeaux, inherited a widely admired scientific library from his father Jean D'Espagnet , a friend of François Viètes (1540–1603), which also contained the works of Vietnam that were very difficult to obtain at the time . This is where the 21-year-old Fermat's career as a mathematician began, after having thoroughly studied Vieta and the ancient mathematicians Euclid and Apollonios von Perge , with his work on maxima and minima and tangents. Around the same time (1629) he reconstructed the two books of Apollonios on the flat places (De locis planis). D'Espagnet and Fermat became friends for life.

### Rise to the Noblesse de Robe

Street side of Fermat's birthplace, wall and gate added in the 18th century; in the background the Fermat tower from the 15th century belonging to the house

Fermat's father died on June 20, 1628. From that moment on, Fermat was a rich man. In order to fulfill the family plan of advancement to the noblesse de robe , however, he first had to complete his four-year practice as a lawyer at one of the highest courts - in Bordeaux - required by a royal decree. On December 29, 1630 he bought the office of conseiller au parlement de Toulouse et commissaire aux requêtes from the widow of the predecessor Pierre de Carriere and died of the plague for the enormous sum of 43,500  livres (a farm laborer earned about 100 livres in one year) was sworn in on May 14, 1631. At the same time, Fermat was raised to the nobility ( noblesse de robe ) , received the title of écuyer (squire, junker) and the right to use the de in front of his name, which he never made use of.

### Marriage to Louise de Long

Signatures of Pierre Fermat and Louise de Long as well as some of the numerous witnesses under the marriage contract of February 18, 1631

During this period, he married Louise de Long, daughter of the influential conseiller Clement de Long and fourth cousin on his mother's side, on June 1, 1631 in the chapel of the prévôté of the Saint-Etienne cathedral in Toulouse. The bride was still very young, born on July 4, 1615, and therefore not quite 16 years old at the time of the wedding. With her he had eight children, five of whom reached adulthood: Clément-Samuel, Jean, Claire, Catherine and Louise. Samuel (he himself only used the biblical name of his godfather, which is preferred by the Huguenots) also became a judge at the parlement de Toulouse, Jean made a career as a clergyman and became archidiacre ( vicar general ) of Fimarcon, and Claire married Michel de Melet, conseiller au parlement de Toulouse . Catherine and Louise entered the Order of the Terzian Sisters of St. Francis in Toulouse.

Saint Etienne Cathedral in Toulouse; the associated Prévôté and its chapel had to give way to a street in the 19th century.

### The Parlement de Toulouse

The chambre aux requêtes was the lowest chamber of the parlement and did not belong to the actual cour. In principle , the French parlements were pure appellate courts, which decided in the last instance after subordinate courts had already ruled. In exceptional cases, especially when high-ranking personalities sought civil proceedings, this principle could be deviated from. The chambre aux requêtes decided on the admission (committimus) of such a request and, if so, assigned the procedure to one of the two chambres des enquêtes . It did not enable advancement to the higher chambers of the parlement. Therefore, Fermat sold this office on December 4, 1637 to Pierre de Caumeil and acquired the office of the late Pierre de Raynaldy am cour and was registered in it on January 16, 1638. He held this office until his death. He was first a member of the first of the two chambres des enquêtes, in which civil cases were decided in the last instance by written procedure. The further rise to the chambre criminelle (la Tournelle) and the Grand'Chambre (the political chamber) then took place according to the principle of anciency .

### Mersennes Republique des Lettres

Andrew Wiles visits Beaumont-de-Lomagne on October 28, 1995 and poses in front of the Fermat monument there, after he had been awarded the
Prix ​​Fermat by the Université Paul Sabatier in Toulouse the day before .

In April 1636, Fermat's childhood friend Pierre de Carcavi moved from Toulouse to Paris and immediately made contact with the famous abbot Marin Mersenne (1588–1648). A few days later Mersenne wrote a (unfortunately lost) letter to Fermat and he replied on April 26, 1636. This began a lively correspondence between Fermat and scholars from many European countries, which Mersenne had for many years (and Carcavi after his death in 1648). Only some of these letters have survived, many of them only in copies from Mersenne. At the end of 1637, Fermat's famous dispute with René Descartes (1596–1650) over the methods of the two scholars for calculating maxima and minima as well as tangents to algebraic curves flared up on October 11, 1638 with a letter from Descartes to Fermat ended.

Fermat's great discoveries in number theory , which have established his fame to this day, fall in the years 1638 to 1643. Fermat wrote his momentous marginal note (cubum autem ...) next to the eighth task of the second book of the Arithmetic of Diophantos of Alexandria in the year 1641 (as in 1638). Among his contemporaries, however, only Bernard Frénicle de Bessy (1605? –1675) was interested in Fermat's number theory research. André Weil reports on them in his masterpiece Number Theory.

### Homo politicus

Toulouse, No. 11 rue Saint-Remesy. Whether Fermat is actually in No. 11 is not secured, but the road is.

Respected mathematicians characterize Fermat's professional life as calm and uneventful as well as socially withdrawn. He himself is described as “meek”, “reserved”, “if not shy at all”. However, these allegations contradict the evidence. For an entire decade, from 1644 to 1653, almost no correspondence with mathematical content has been preserved from Fermat. It cannot be determined whether he even found time to continue his mathematical research during these years. But it can be proven that he was severely stressed by the political events and his role in them.

There are no indications that Fermat ever aspired to a political career. For this he probably lacked the appropriate ambition and the necessary ruthlessness. But he was very much a homo politicus who energetically and courageously took on responsibility. He had ample opportunity to do so. It turned out that Fermat had a special negotiating skill, which he demonstrated when it came down to mediating between the interests of conflicting parties - a talent that he had probably inherited from his father Dominique.

### Travel on duty

Toulouse, rue Saint-Remesy, where Fermat lived

"Fermat finally died", writes André Weil , "without ever venturing further away from home than to Bordeaux." On November 28, 1646, the Grand'Chambre had for protectionist reasons forbade the dyers of Nîmes by decree, the indigofera tinctoria to buy and use for dyeing. The existing supplies should be delivered. Instead, the teinturiers should use the isatis tinctoria ( woad ) grown in the Lauragais . When the King's Attorney General found that the dyers of Nîmes were permanently disregarding this prohibition, the cour commissioned the conseiller Fermat in May 1647 to go to Nîmes, 300 km east of Toulouse, in order to enforce the Grand'Chambre resolution procure. That meant a week-long trip on horseback there and back. How Fermat succeeded in asserting himself among the dyers of Nîmes is not known. In any case, it wasn't a job that would have been entrusted to a meek, reserved, or even shy man. In his letters to Mersenne, Fermat occasionally reports that he was sent on various missions far from Toulouse.

### The years of the Fronde

Pierre de Fermat by Jean Bertrand Loubens (1848–1913), marble statue on the former Faculté des Sciences, Toulouse

The years of the Fronde (1648–1653) followed. Also the parlement de Toulouse participated in the fronde parlementaire (1648-1649), in which for most it parlements to the extreme increase in taxes, especially to the waist , to left, the withdrawal of which they demanded from the Crown, and the brutal and illegal methods of collecting them:

Andrew Wiles on October 28, 1995 on the platform of the tower belonging to Fermat's birthplace

"On July 18, 1648, the united chambers of the parlement decided to make representations to the sovereign and to order provisionally that the tax collectors (partisans) were forbidden to collect the waist by force of arms and by the compulsory billeting of soldiers."

Fermat had also uncovered the "crooked tour" (voies obliques) of the backdated receipts, which enabled the partisans to embezzle the monies due to the king for their profit. Fermat, who had good personal contact with Marin Cureau de la Chambre (1594–1669), personal physician to the king and friend of Chancellor Pierre Séguier (1588–1672), due to shared scientific interests (theory of light) , was commissioned to find a corresponding one To send a letter to Séguier via de la Chambre. He sent this letter to de la Chambre on August 18, 1648 , in which Fermat submitted proposals on behalf of the parlement as to how the taxes could be raised in other ways. It had no effect. Other " parlements proposed remedies in their remonstrances that were completely illusory."

When the fronde des Princes (1650–1652) broke out in Languedoc at the end of 1650 , violent clashes quickly broke out between the estates (états) of the province, who had joined the uprising of the Prince de Condé , and the parlement de Toulouse, which was loyal to the king. The estates no longer recognized the jurisdiction of the parlement and established independent assemblies without authorization from the crown. Finally, in December 1651, it was agreed to form two eight-person commissions to negotiate to settle the conflict. The parliamentary commission consisted of the premier président, two présidents à mortier and five experienced conseillers, one of whom was Fermat. The commissions negotiated for a whole year until legal peace was restored.

### Commitment to Beaumont-de-Lomagne

Beaumont-de-Lomagne from the south-east. The royal troops had their camp on the opposite hill.

From October 1651 to August 1652, Fermat's birthplace Beaumont-de-Lomagne was occupied and plundered by Condé's troops under the military command of Jacques de Guyonnet, a conseiller au parlement de Bordeaux . They deposed the city's consuls and collected the taxes due to the king. When, in July, the outnumbered troops of the king, led by Count Savaillant, pitched their camp on a hill above Beaumont and threatened to storm the city by force of arms, the citizens turned to the two lawyers Pierre de, who came from Beaumont, to mediate Fermat and Abraham de Toureil (Tourreil), procureur général au parlement de Toulouse (Attorney General). Led by Fermat and accompanied by the consuls Breville and Cirol, they went to the camp of the royal troops on July 30, 1652, and negotiated that Condé's troops would be granted free retreat and that the city would be saved from destruction. To make matters worse, the completely looted Beaumont was sentenced to a fine of 15,000 livres, which the citizens could not have raised. The sum was paid in equal parts by Fermat, Toureil and Fermat's friend Jean-Georges de Saliné, Seigneur de Roujos.

### Plague disease

This relief shows the coats of arms in the second line and below the names of the four consuls from 1617 of Beaumont-de-Lomagne, and in third place that of the godfather of the mathematician Pierre de Fermat, brother of his father Dominique.

In 1652 Fermat moved to the chambre criminelle (la Tournelle) des parlement de Toulouse for reasons of anciency . In the autumn of this year, the last great plague epidemic broke out in Toulouse, which did not end until July 1653 and which killed around 4,000 people (one tenth of the population of Toulouse) in the city. In May 1653 a friend, the philosopher Bernard Medon , reported Fermat's death in a letter to Nicolaas Heinsius the Elder (1620–1681):

"Fato functus est maximus Fermatius"

"The great Fermat has died [literally: has completed his life goal]"

however revoked this in the next letter:

"Priori monueram te de morte Fermatii, vivid adhuc, nec desperatur de ejus salute, quamvis paulo ante conclamata"

“I recently reported to you about Fermat's death. He is alive, and you no longer despair of saving your life, as recently lamented so much. "

Fermat survived, probably because of good medical / surgical care and robust health, the bubonic plague , to which around half of those affected fell victim. However, there are many indications that his health has been weakened since that time. Fermat's life, like most French people in the first half of the 17th century, was repeatedly affected by the bubonic plague. The possibility of buying himself the office of a conseiller au parlement de Toulouse in 1630 , probably the death of his brother Clément, who was barely 20 years old, in 1631/32 and his own life-threatening illness in 1653 were all connected with the plague epidemics.

### Pascal and the calculation of happiness

Fermat sculpture by Alexandre Falguières, here still en pierre, again in bronze since 2013 (the German occupation had melted the original bronze sculpture in 1943)

After his recovery, in November 1654 he routinely moved to the Grand'Chambre, the highest and political chamber of the parlement de Toulouse (annually two judges were exchanged between the Grand'Chambre and the Tournelle , they were therefore considered a single, but two-part chamber ). Between July and October 1654, the famous exchange of letters between Blaise Pascal (1623–1662) and Fermat took place on the fair distribution of the stakes in a game of chance if the game was stopped prematurely. This correspondence is considered to be a milestone in the early history of probability theory . In November 1655, Fermat was again delegated to the Chambre de l'Edit for two sessions (1655/56 and 1656/57) in Castres.

### The Chambre de l'Edit in Castres

Hôtel des Reformed lawyer Rozel in Castres (17th century), one of the few remaining evidence of the city's most glorious times

His frequent secondment as a Catholic judge to the Chambre de l'Edit (de Nantes) in Castres in the sessions (from November to August) 1638/39, 1644/45, 1645/46, 1648 / played an important role in his professional life. 49, 1649/50, 1655/56, 1656/57, 1663/64 and 1664/65. No other Catholic judge was sent to Castres so often.

Fermat succeeded in doing this only by taking advantage of the influential protection of Marin Cureau de la Chambre, the king's personal physician and close friend of Chancellor Pierre Séguier. Fermat and de la Chambre shared an interest in the physical theory of light. And Séguier decided in the last instance on the delegation of the Catholic Conseillers to the Chambre de l'Edit in Castres. This chamber was finally established by Henry IV in 1598 in the Huguenot stronghold of Castres and was made up of ten Reformed and ten Catholic judges each. All civil and criminal processes in which persons of both denominations were involved were carried out in it.

Fermat was an enlightened and tolerant Catholic, with a clear sympathy for the Reformed religion. His mother came from a noble Huguenot family from Montauban, he himself received his much-praised classical education at the Reformed collège de Navarre in Montauban and had several close friends among the Reformed judges and lawyers in Castres, including Pierre Saporta (1613–1685?) And Jacques de Ranchin (1620-1692).

### The reformed academy of scholars

In Castres the educated Huguenots had founded a scientific and literary academy in 1648, in which Catholics were not admitted as members. In it, Ranchin von Fermat and his son Samuel read pious and frivolous Latin poems. For his part, Fermat dedicated his critical remarks to Ranchin on the work of the Greek historian Polyainus (2nd century AD), thereby giving a taste of his mastery of Greek philology. In 1664, Saporta published one of the few texts that appeared in print during Fermat's lifetime. It is a short text in which he explains a passage from a letter from Synesius of Cyrene , where he describes a “hydroscope” or “baryllon”. Fermat calls it an "areometer". Fermat's youngest daughter Louise was born in Castres, and his younger son Jean was canon in Castres when his father died there in January 1665.

### The source of wealth

Fermat monument in front of the market hall in Beaumont, sculpture by Alexandre Falguières (1831–1900)

Fermat used to spend his parliamentary holidays (September / October) in his birthplace Beaumont-de-Lomagne, in the vicinity of which he owned six larger estates that were inherited from his father (as well as some vineyards). He had all leased them on a métayage basis, which means that at harvest time he received half of the fruit produced by the tenant. Fermat and his family lived from selling them, but his income from his work as a conseiller was rather low. From an economic point of view, getting the office of conseiller in one of the parlements was bad business. The gain, however, consisted in the rise to the noblesse de robe, in the privileges associated with the office, in the high social prestige and, upon reaching the Grand'Chambre, in participation in political power.

### Correspondence with Wallis, Brouncker and van Schooten

In 1656 Sir Kenelm Digby (1603-1665) visited Fermat, either in Castres or in Beaumont. Through Digby's agency, who stayed mainly in Paris until 1660, a lively mathematical correspondence developed from January 1657 to June 1658, to which John Wallis (1616–1703), William Lord Viscount Brouncker (1620? –1684) , Frans van Schooten (1615? –1660) and Frénicle de Bessy involved. In the end, it turned out to be rather unsatisfactory for Fermat, because the gentlemen (with the exception of Frénicle) neither understood nor knew how to appreciate his number theoretic challenges.

### Fermat and the Delpoy case

Death sentence against the priest Raymond Delpoy (first of four pages) of January 9, 1658, rapporteur: Fermat

After his return in November 1657 from Castres in the chambre criminelle (la Tournelle) in Toulouse Fermat was rapporteur (rapporteur) involved in a spectacular miscarriage of justice, of Fermat's professional life until his death overshadowed. This case, known only from a single reference (which falsifies the facts) in a letter from the “notorious liar” Sir Kenelm Digby to John Wallis on February 6, 1658, was only cleared up in 2007. At the instigation of the chief judge (premier président) of the parlement de Toulouse, Gaspard de Fieubet (1622–1668), the Catholic priest Raymond Delpoy was sentenced to death on January 9, 1658 by the united chambers of la Tournelle and Grand'Chambre because he his confessor Françoise de Lacombe, a 14-year-old Huguenot who was being re-educated into a “good Catholic” in the maison des nouvelles converties run by the infamous Madame de Mondonville, escaped from this very house and helped her with his brother, the Doctor Pierre Delpoy, in Limoux, about 100 kilometers away. The priest was hanged, his body burned on a pyre, and his ashes scattered to the wind. The otherwise extremely hardworking Fermat could hardly work as a judge for several months (not even on the proof of an uninteresting theorem by Wallis), lost interest in his office at the cour and wanted to bequeath it to his son Samuel, which failed because he had not yet Was 25 years old. Since then he has been enemies with the premier president Gaspard de Fieubet.

### Assessment by Bazin de Bezons

Fermat with Muse, by Théophile Eugène Victor Barrau (1848–1913), in the Salle des Illustres des Capitole, Toulouse

In a decree of September 1663, the minister Jean-Baptiste Colbert had asked the intendants (the governors of the French provinces) to deliver an assessment of all judges and other high judicial officials in all parlements falling under their jurisdiction . Claude Bazin de Bezons (1617–1684), director of the Languedoc , was responsible for Toulouse . He traveled from his seat in Montpellier to Toulouse in November 1663 (Fermat had just left for Castres) and delivered his laconic brief report on the presidents and the judges, a total of only 41 people. Colbert was not very enthusiastic and demanded more detailed representations from some of the directors, he let Bazins pass. Here is a typical example:

«Rabaudy, de bonne maison, mais homme de rien. »

("Rabaudy, from a good family, but a zero.") About the controversial first president:

«Fieubet, premier président, affectionne la justice et le service de roy, a assez de talent pour parler en public; corn peu riche et rompu dans le monde; a des amis dans le parlement, mai n'y a pas une estime ni approbation générale, et a une assez forte cabale contre luy. »

“Fieubet, First President, values ​​justice and the service of the King, and has a marked ability to speak in public; but is not very rich and gets lost in his social activities [literally: is torn in the world]; he has friends in parliament, but he does not enjoy general esteem and recognition, and there is a very strong intrigue against him. "

«Fermat, homme de beaucoup d'érudition, a commerce de tous costés avec les sçavants, mais assez intéressé; n'est pas trop bon rapporteur et est confus, n'est pas des amys du premier président. »

“Fermat, a man of great learning, has dealings with scholars everywhere, but rather selfishly; he is not a very good reporter and is confused, he is not one of the friends of the first president. "

- : Correspondance administrative sous le règne de Louis XIV.

Henri Gilles (1921–2012), Professor of Legal History at the Université de Toulouse 1, Fermat researcher

This judgment led several biographers of Fermat (most notably Mahoney) to consider him a poor lawyer. Bazin’s judgment is not surprising, as he based it on information from Fermat’s intimate enemy Fieubet; He could not meet Fermat himself because of his absence. Henri Gilles (1921–2012), legal historian at the Université Toulouse I, researched Fermat's work as a judge for decades in the Archives départementales de la Haute Garonne and published an essay on it as early as 1965, eight years before the publication of the first edition of Mahoney's work. In it he writes:

“This disparaging assessment undoubtedly betrays the antipathy that Prime Minister Gaspard de Fieubet feels with regard to Fermat, of which the quoted note assures us that Fermat is not one of his friends. This does not seem to be justified; Throughout his life, Fermat cultivated a very clear style, and the language of the reports he has written need not shy away from comparison with that of his colleagues. "

### The law of refraction and the principle of the shortest path

«  Le dousième du mois de Janvier 1665 décéda ayant reçu tous les sacremens messire Pierre de Fermat conseiller du Roy en son parlement de Tolose et commissaire en la chambre de Ledit séant à Castres et feust ensevelly le tretsième dans l'eglise des révérends Pères de Stévérends Dominique ou les messieures du Vénérable Chapitre ont faict loffice - Cabrier pbre  »(Archives de Castres, GG II)

After Descartes' mysterious death on February 11, 1650, Claude Clerselier (1614–1684), the editor and translator of Descartes' writings and his ardent admirer, began to prepare Descartes' letters for publication. At the beginning of 1657 Clerselier turned to Fermat to resume his discussion with Descartes of 1637 about Snellius's law of refraction , probably in the hope of persuading Fermat to admit the superiority of Descartes' position. Instead, however, it became an exchange of letters that lasted until May 21, 1660, in which Clerselier forced Fermat through constant objections to repeatedly improve his argumentation, until Fermat finally succeeded in producing a mathematically sound proof that one was Snellius' law of refraction can derive from Fermat's principle of the shortest path in time. Fermat was not the discoverer of the law of refraction. The significance of his last great mathematical achievement lies in his application of the (still integral-free) calculus of variations , in the history of which it occupies an important place. Then he allowed his beloved "geometry" to "fall into a deep sleep".

### Preparation for the end and death

This plaque marks the spot on Place Jean-Jaures in Castres where, in the 17th century, the chapel was located where Pierre de Fermat was (temporarily) buried the day after his death. This chapel was later demolished to make way for Place Jean-Jaures.
Musée des Augustins in Toulouse, before the French Revolution Augustinian monastery, in whose church at that time the tomb of the Fermats

In a letter of July 25, 1660 to Blaise Pascal, who was also sickly like Fermat, he suggested meeting Pascal halfway between Clermont-Ferrand and Toulouse, “parce que ma santé n'est guere plus forte que la votre "(" Because my health is not much better than yours "). If Pascal is not ready and expects Fermat to cover the whole distance between the two cities, around 380 km, “vous courrez hazard de me voir chez vous et d'y avoir deux malades en même temps” (“you would run the risk to see me with you and to have two sick people there at the same time ”). In the same year, on March 4th, Fermat had written his will and in it appointed his older son Samuel as the universal heir. He added to this will on September 13, 1664, apparently in anticipation of his not distant death, with a codicil in which he obliged his son Samuel to pay his mother Louise de Long 32,000 livres from the inheritance. She could use this sum, because she outlived her husband by at least 25 years; their signature can be found under a document dated October 3, 1690. The codicil begins with the words: “Je soubsigné éstam [étant] incommodé d'une maladie qui pourroit avoir mauvaises suittes…” (“I, the undersigned, suffer from an illness that could have dire consequences ”). Once again he continued his work at the Chambre de l'Édit in Castres and fulfilled his judicial duties to the end. His signature under a resolution is found for the last time on January 5, 1665, and his last report was read by him in the Chamber on January 9, but no longer signed. He dies in Castres on January 12, 1665 , after receiving the sacraments and until the end with clear consciousness.

### Fermat's epitaph

"Piæ memoriæ dom Petri de Fermat senatoris Tolosani qui literarum politiorum pluriumque linguarum, et matheseos ac philosophiæ peritissimus, ita jurisprudentiam calluit. - Ita iudicis munere functus est. Ut eius ad hoc unum collecta crederetur ingenii vis, licet in tot arduas speculationis divisa. - Vir ostentationis expers, suas lucubrationes typis mandari non curans, et egregiorum operum neglectu adhuc maior quam partu, præclara sui legit in aliorum libris elogia nec intumuit. Nunc autem, quod ipsius virtutes sperare sinunt dum æternam veritatem contemplari gaudet, cœlesti radio maxima et minima dimensus, e tumulo quemlibet affari videtur, hoc aureo christiani doctoris monito - Vis scire quiddam quod juvet? nesciri ama. IF. XII. IAN. M.DC.LXV. AET. ON. LVII "

He was buried the next day - initially - in the Jacobin chapel in Castres. In the same year, his sons transfer their father's remains to the family grave in the church of the Augustinian monastery in Toulouse, which Samuel had established a few years earlier. This tomb was destroyed during the French Revolution. Today the former monastery is a pure art museum, but it has a treasure in its collection (in a warehouse 75 km northwest of Toulouse): the epitaph Pierre de Fermats from the destroyed family mausoleum that his son had made by a stonemason in 1665. His last line is written in detail:

"Obiit duodecimo Ianuarii mille sescenti sexaginta quinque aetate annorum quinquaginta septem"

"He passed away on January 12, 1665 at the age of 57."

## Post fame

In his later years, Fermat, who had not had any of his mathematical findings printed, began to worry about what, above all, his number-theoretical findings would be preserved for posterity. In his last letter to Huygens of August 1659, he again lists his most important findings in this area and then concludes with the words:

“That is, in a nutshell, the report on my dreams about numbers. ... And maybe posterity will thank me for telling them that the ancients did not know everything. This communication will be seen as the handing over of the torch to the young in the spirit of those who come after me, as the Grand Chancellor of England [Bacon] put it, and I add, following his opinion and motto: Multi pertransibunt et augebitur scientia ( Many will go there, but science is growing). "

When, on February 9, 1665, a Fermat obituary written by Carcavi appeared in the Journal des Sçavans, it - typically - only listed examples of calculus and analytical geometry in addition to Fermat's philological achievements in mathematics. The math historian Keith Devlin writes :

“When Pierre de Fermat died on January 12, 1665, he was one of the most famous mathematicians in Europe. Although today his name is invariably associated with number theory , much of his work in this field was so far ahead of its time that he was better acquainted with his contemporaries through his research in coordinate geometry (which he invented independently of Descartes), through calculus (which Newton and Leibniz completed) and by probability theory (which was essentially founded by Fermat and Pascal). "

Fermat's number theoretic findings were almost completely forgotten for a long time. It was not until 1730 that Leonhard Euler took up Fermat's “torch” of number theory again by rediscovering and continuing it. Today Fermat is considered the most important mathematician of the first half of the seventeenth century, not least because he founded modern number theory. Jean Dieudonné writes:

“Fermat, undoubtedly the most profound mathematician of the seventeenth century, created the foundations of probability theory with Pascal and discovered, before Descartes, the coordinate method. He was the first to introduce a general method of determining tangents to plane curves; but it was above all number theory in which his genius was revealed. "

The Concise Oxford Dictionary of Mathematics only notes briefly:

"Fermat, Pierre de (1601-65). Leading mathematician of the first half of the seventeenth century. "

## Contributions to mathematics

Fermat, like most scientists of his time, did not deal full-time with mathematics . Rather, he was a busy and committed judge at the parlement de Toulouse. So his influence was limited to the correspondence with many important scholars of his time. Correspondence has been received with (alphabetically) Jean de Beaugrand , Pierre Brûlart de Saint-Martin , Claude Clerselier , René Descartes , Bernard Frénicle de Bessy , Pierre Gassendi , Christiaan Huygens , Étienne Pascal , Blaise Pascal , Gilles de Roberval and John Wallis . Other correspondence, for example with Galileo Galilei and Evangelista Torricelli , went indirectly through Marin Mersenne or Pierre de Carcavi . An important source is the edition of his estate made by his son, including the arithmetic of Diophantos of Alexandria , which he commented on . He made groundbreaking contributions to differential calculus , analytical geometry , number theory , probability theory and the calculus of variations . He often only communicated his findings in the form of “challenges” (défis) , that is, he only gave the result, not the solution.

The following are named after Fermat:

• The Fermat's principle is a principle of variation of the optical "light makes its way ever so that it travels it in the shortest time." From this, the guided reflection law and Snell's law from.
• As Fermat numbers numbers are of the form F n  = 2 2 n  + 1, with n ∈  designated. Fermat hypothesized in 1637 that all Fermat numbers are prime numbers . However, this was refuted by Euler in 1732 , who showed that the sixth Fermat number F 5 is divisible by 641. Apart from the first five (3, 5, 17, 257, 65537), no other Fermat prime numbers are known, and one assumes that there are none.
• The Fermat two-square theorem as follows: An odd prime number is exactly the sum of two squares, when viewed from the form is:${\ displaystyle p}$${\ displaystyle p = 4n + 1}$
${\ displaystyle \ left (\ exists (x, y) \ in \ mathbb {N} ^ {2} \ quad p = x ^ {2} + y ^ {2} \ right) \; \; \ Leftrightarrow \; \; p \ equiv 1 {\ pmod {4}},}$
and this representation is unambiguous except for the order of the summands.
The first proof of this theorem goes back to Euler . The two smallest prime numbers with this property are 5 = 1 2  + 2 2 and 13 = 2 2  + 3 2 .
• Little Fermat's theorem : For every prime numberwe have:${\ displaystyle p}$
${\ displaystyle a ^ {p} \ equiv a \ (\ mathrm {mod} \ p)}$for everyone .${\ displaystyle a \ in \ mathbb {N}}$
The Fermat's prime number test is based on this theorem . Also in this case, the first place received evidence at Euler.
Fermat proved more than what is mostly, and also here, quoted as Fermat's Little Theorem, namely:
Fermat's theorem : Let it bea positive integer and a prime number thatdoes not divide. Then there are positive integersfor which ${\ displaystyle a}$${\ displaystyle p}$${\ displaystyle a}$${\ displaystyle n}$
${\ displaystyle a ^ {n} \ equiv 1 \ (\ mathrm {mod} \ p)}$
holds, and let it be the smallest such number. Then and ${\ displaystyle d}$${\ displaystyle d \ mid p-1}$
${\ displaystyle a ^ {n} \ equiv 1 \ (\ mathrm {mod} \ p) \ iff d \ mid n}$.
In particular,
${\ displaystyle a ^ {p-1} \ equiv 1 \ (\ mathrm {mod} \ p).}$
This variant of his theorem enabled him to find the prime divisor of the Mersenne number and thus to prove that it is not a perfect number .${\ displaystyle 223}$ ${\ displaystyle 2 ^ {37} -1 = 137438953471}$${\ displaystyle 2 ^ {36} (2 ^ {37} -1)}$
a n  + b n  = c n
with a, b, c ∈  for no natural number n> 2 is fulfilled. So there are no analogues to the Pythagorean triples for the third or higher powers. This sentence became famous because Fermat claimed in a marginal note of his copy of Diophant's Arithmetic that he had found a “truly wonderful” proof of this, but for which “there was not enough space in the margin”:

“Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et generaliter nullam in infinitum ultra quadratum potestatem in duas ejusdem nominis fas est dividere: cujus rei demonstrationem mirabilem sane detexi. Hanc marginis exiguitas non caperet. "

“However, it is not possible to divide a cube into 2 cubes, or a biquadrat into 2 biquadrates and generally one power, higher than the second, into 2 powers with the same exponent: I have discovered a truly wonderful proof of this, but this one is Edge here too narrow to hold it. "

- Pierre de Fermat
The case n  = 4 was proven by Fermat elsewhere (with his method of infinite descent ), further cases later by other mathematicians. In its generality, the statement remained one of the most famous unsolved problems in mathematics until the mid-1990s . It was not until September 1994 (published in May 1995 with a contribution by Richard Taylor ) that the British mathematician Andrew Wiles succeeded in proving Fermat's conjecture. This is why it is also known as the Fermat / Wiles theorem or the Wiles and Taylor theorem. Wiles was awarded the Abel Prize for proof in 2016 .

## Honors

In honor of Fermat, the Paul Sabatier University in Toulouse has awarded the Prix ​​Fermat every two years since 1989 . Andrew Wiles also received this award on October 27, 1995. There is also a Pierre Fermat Medal from the Académie des sciences inscriptions et belles lettres de Toulouse , which, for example, Linus Pauling received in 1957 (as the sixth prize winner ever).

The lunar crater Fermat was named after him by the IAU in 1935 , as was the asteroid (12007) Fermat in 2000 .

## Trivia

In an episode of the Spaceship Enterprise series , Captain Picard admits he's trying to prove Fermat's theorem. At the end of the episode he literally speaks of the “... Fermat equation. A riddle that we will probably never solve. ”Since this episode (2nd season, 12th episode Hotel Royale ) was filmed in 1989, the solution was still unknown at the time. A correction was subsequently suggested in the 1995 episode DS9 : Facets .

## Fonts

During Fermat's lifetime, only one treatise by Fermat on integration by Antoine de Lalouvère was published without naming the author as an appendix to his Veterum geometria promota in septem de cycloide libris (1660).

## Individual evidence

1. Klaus Barner: How old did Fermat become? In: NTM International Journal for the History and Ethics of Natural Sciences, Technology and Medicine, New Series 9, 2001, pp. 209–228 (English; Zentralblatt review ).
2. ^ Wikipedia discussion on "Fermat's birthday".
3. ^ Registres paroissiaux de Beaumont, actes des baptêmes. Piere, fils de Dominique Fermat, bourgoys et segont consul de la ville de Beaumont, a esté baptisé le 20 aoust 1601, parrin Piere Fermat, marchant et frère dudit Dominique, marrine Jehanne Cazanove. Par moy DUMAS, vicario. From the second half of 1607 to 1611 inclusive, all Beaumont baptismal registers have been lost.
4. ^ Klaus Barner: News on Fermat's date of birth. In: Mitteilungen der DMV  15, 2007, pp. 12-14.
5. ^ Wikipedia discussion on the subject of "leather".
6. Pierre Gairin: Pierre Fermat et ses ascendants. Publié à compte d'auteur, Beaumont-de-Lomagne 2001, p. 12 (French). Dominique Fermat was elected second consul for 1601 and first consul of Beaumont for 1605 and 1613.

“In the Middle Ages, the name Consul appeared as the title of the ruling officials of the city communes in Italy and southern France, first attested in Pisa in 1081. The consuls, mostly elected for one year, directed administration, the military and the judiciary, but were controlled by the people's assembly and the city council. "

8. Joseph Bessodès: Beaumont-de-Lomagne: son histoire des origines au début du XIX e siècle. Syndicat d'Initiative, Beaumont-de-Lomagne 1993, p. 97 (French).
9. ^ Horst Heintze in François Rabelais : Gargantua and Pantagruel. Insel Verlag, Frankfurt am Main 2003, p. 10. The humanist Rabelais resigned from the Franciscan order for this reason.
10. ^ Michael Sean Mahoney: The Mathematical Career of Pierre de Fermat 1601–1665. Princeton University Press, Princeton, New Jersey, 1973, p. 15.
11. Immo Eberl : The Cistercians. History of an order. Scientific Book Society, Darmstadt 2002, p. 67.
12. Pierre Gairin: Pierre Fermat et ses ascendants. Publié à compte d'auteur, Beaumont-de-Lomagne 2001, p. 71 (French).
13. Georges Bourbon: Notice historique sur le collége de Montauban, depuis sa fondation jusqu'en 1792. Bulletin archéologique et historique 4, 1876, pp. 185-214 (French).
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17. Bibliothèque d'Université Toulouse, ms 33, anc. 28, fol. 396. (Fermat, aggreg. Baccall. In Civili. Discretus vir Petrus Fermat, diocesis Montisalbani, baccall. Apud Universitatis Aurelian. In Jure Civili, fuit aggregatus baccall. In eadem Facultate sub domino Maran 2, die prima mensis maii, 1631).
18. Jean-Eugène Bimbenet: Histoire, de lois d'Orléans. Gatineau, Orléans 1853 (French).
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27. ^ Roland Mousnier: Les institutions de la France sous la monarchie absolue 1598–1789. Volume 2, Les organes de l'Etat et la Société. Presses Universitaires de France, Paris 1980, ISBN 2-13-036307-5 , p. 37 (French).
28. ^ Pierre Goubert: The French Peasantry in the Seventeenth Century. Cambridge University Press, Cambridge 1986, p. 111 (English).
29. The marriage contract was signed on February 18, 1631 in the house of a notary in Toulouse (Archives départementales de la Haute Garonne, Minutier, 3E 6070, 2e partie, ff. 376–378). Pierre Carcavi, with whom Fermat had apparently been friends for a long time, also signed as one of the witnesses.
30. ^ Michael Sean Mahoney: The Mathematical Career of Pierre de Fermat 1601–1665. 1994, pp. 170-195 (English).
31. ^ Ivo Schneider : Descartes' discussion of Fermat's extreme value method - a piece of the history of ideas in mathematics. In: Archive for History of Exact Sciences  7, 1971, pp. 354-374 ( Zentralblatt-Autorreferat ).
32. Hal Hellman: Great Feuds in Mathematics: Ten of The Liveliest Disputes ever. John Wiley & Sons, Hoboken, NJ, 2006, pp. 26-44.
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35. ^ Israel Kleiner: Excursions in the History of Mathematics. Birkhäuser, Basel 2012. Chapter 2. Fermat: The Founder of Modern Number Theory. Pp. 31-45 (English). This chapter gives a not too brief but also not too extensive overview of Fermat's most important results in number theory, unfortunately again only in English.
36. Roshdi Rashed : Histoire de l'analyse diophantienne classique. D'Abū Kāmil à Fermat. De Gruyter, Berlin 2013 (French).
37. Eric Temple (ET) Bell: Men of Mathematics. The Lives and Achievements of the Great Mathematicians from Zeno to Poincaré. Simon & Schuster, New York 1937. Chapter four. The Prince of Amateurs: Fermat. Pp. 57-59 (English).
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41. Pierre Gairin: Pierre Fermat et ses ascendants. Publié à compte d'auteur, Beaumont-de-Lomagne 2001, pp. 8-13 (French). In this publication, Gairin gave, among other things, a biography of Fermat's father Dominique based on numerous documents, which also demonstrates his outstanding negotiating skills.
42. ^ André Weil: Number theory. A walk through history from Hammurabi to Legendre. Birkhäuser, Basel 1992, p. 40.
43. ^ Archives départementales de la Haute Garonne. Vol. 686, fol. 72.
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