Pierre de Fermat

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Pierre de Fermat (; between 31 October and 6 December 1607 – 12 January 1665) was a French
mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...
who is given credit for early developments that led to infinitesimal calculus, including his technique of adequality. In particular, he is recognized for his discovery of an original method of finding the greatest and the smallest ordinates of curved lines, which is analogous to that of differential calculus, then unknown, and his research into
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative intege ...
. He made notable contributions to
analytic geometry In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry using a coordinate system. This contrasts with synthetic geometry. Analytic geometry is used in physics and engineerin ...
, probability, and
optics Optics is the branch of physics that studies the behaviour and properties of light, including its interactions with matter and the construction of optical instruments, instruments that use or Photodetector, detect it. Optics usually describes t ...
. He is best known for his Fermat's principle for light propagation and his Fermat's Last Theorem in
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative intege ...
, which he described in a note at the margin of a copy of Diophantus' '' Arithmetica''. He was also a lawyer at the '' Parlement'' of
Toulouse Toulouse ( , ; oc, Tolosa ) is the Prefectures in France, prefecture of the Departments of France, French department of Haute-Garonne and of the larger Regions of France, region of Occitania (administrative region), Occitania. The city is on t ...
,
France France (), officially the French Republic ( ), is a country primarily located in Western Europe. It also comprises of Overseas France, overseas regions and territories in the Americas and the Atlantic Ocean, Atlantic, Pacific Ocean, Pac ...
.

# Biography

Fermat was born in 1607 in Beaumont-de-Lomagne, France—the late 15th-century mansion where Fermat was born is now a museum. He was from Gascony, where his father, Dominique Fermat, was a wealthy leather merchant and served three one-year terms as one of the four consuls of Beaumont-de-Lomagne. His mother was Claire de Long. Pierre had one brother and two sisters and was almost certainly brought up in the town of his birth. He attended the University of Orléans from 1623 and received a bachelor in civil law in 1626, before moving to
Bordeaux Bordeaux ( , ; Gascon language, Gascon oc, Bordèu ; eu, Bordele; it, Bordò; es, Burdeos) is a port city on the river Garonne in the Gironde Departments of France, department, Southwestern France. It is the capital of the Nouvelle-Aquita ...
. In Bordeaux, he began his first serious mathematical researches, and in 1629 he gave a copy of his restoration of Apollonius's '' De Locis Planis'' to one of the mathematicians there. Certainly, in Bordeaux he was in contact with Beaugrand and during this time he produced important work on maxima and minima which he gave to Étienne d'Espagnet who clearly shared mathematical interests with Fermat. There he became much influenced by the work of François Viète. In 1630, he bought the office of a councilor at the Parlement de Toulouse, one of the High Courts of Judicature in France, and was sworn in by the Grand Chambre in May 1631. He held this office for the rest of his life. Fermat thereby became entitled to change his name from Pierre Fermat to Pierre de Fermat. On 1 June 1631, Fermat married Louise de Long, a fourth cousin of his mother Claire de Fermat (née de Long). The Fermats had eight children, five of whom survived to adulthood: Clément-Samuel, Jean, Claire, Catherine, and Louise. Fluent in six languages ( French,
Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally a dialect spoken in the lower Tiber area (then known as Latium) around present-day Rome, but through ...
, Occitan, classical Greek, Italian and Spanish), Fermat was praised for his written verse in several languages and his advice was eagerly sought regarding the emendation of Greek texts. He communicated most of his work in letters to friends, often with little or no proof of his theorems. In some of these letters to his friends, he explored many of the fundamental ideas of calculus before Newton or Leibniz. Fermat was a trained lawyer making mathematics more of a hobby than a profession. Nevertheless, he made important contributions to analytical geometry, probability, number theory and calculus. Secrecy was common in European mathematical circles at the time. This naturally led to priority disputes with contemporaries such as Descartes and Wallis. Anders Hald writes that, "The basis of Fermat's mathematics was the classical Greek treatises combined with Vieta's new algebraic methods."

## Work

Fermat's pioneering work in
analytic geometry In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry using a coordinate system. This contrasts with synthetic geometry. Analytic geometry is used in physics and engineerin ...
(''Methodus ad disquirendam maximam et minimam et de tangentibus linearum curvarum'') was circulated in manuscript form in 1636 (based on results achieved in 1629), predating the publication of Descartes' famous '' La géométrie'' (1637), which exploited the work. This manuscript was published posthumously in 1679 in ''Varia opera mathematica'', as ''Ad Locos Planos et Solidos Isagoge'' (''Introduction to Plane and Solid Loci''). In ''Methodus ad disquirendam maximam et minimam'' and in ''De tangentibus linearum curvarum'', Fermat developed a method ( adequality) for determining maxima, minima, and
tangent In geometry, the tangent line (or simply tangent) to a plane curve at a given Point (geometry), point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitesimal, infinitely ...
s to various curves that was equivalent to differential calculus. In these works, Fermat obtained a technique for finding the centers of gravity of various plane and solid figures, which led to his further work in quadrature. Fermat was the first person known to have evaluated the integral of general power functions. With his method, he was able to reduce this evaluation to the sum of geometric series. The resulting formula was helpful to Newton, and then Leibniz, when they independently developed the fundamental theorem of calculus. In number theory, Fermat studied Pell's equation,
perfect number In number theory, a perfect number is a positive integer that is equal to the sum of its positive divisors, excluding the number itself. For instance, 6 has divisors 1, 2 and 3 (excluding itself), and 1 + 2 + 3 = 6, so 6 is a perfect number. T ...
s, amicable numbers and what would later become Fermat numbers. It was while researching perfect numbers that he discovered Fermat's little theorem. He invented a factorization method— Fermat's factorization method—and popularized the proof by infinite descent, which he used to prove Fermat's right triangle theorem which includes as a corollary Fermat's Last Theorem for the case ''n'' = 4. Fermat developed the two-square theorem, and the polygonal number theorem, which states that each number is a sum of three triangular numbers, four square numbers, five pentagonal numbers, and so on. Although Fermat claimed to have proven all his arithmetic theorems, few records of his proofs have survived. Many mathematicians, including Gauss, doubted several of his claims, especially given the difficulty of some of the problems and the limited mathematical methods available to Fermat. His famous Last Theorem was first discovered by his son in the margin in his father's copy of an edition of Diophantus, and included the statement that the margin was too small to include the proof. It seems that he had not written to Marin Mersenne about it. It was first proven in 1994, by Sir Andrew Wiles, using techniques unavailable to Fermat. Through their correspondence in 1654, Fermat and
Blaise Pascal Blaise Pascal ( , , ; ; 19 June 1623 – 19 August 1662) was a French mathematician, physicist, inventor, philosopher, and Catholic Church, Catholic writer. He was a child prodigy who was educated by his father, a tax collector in Rouen. Pa ...
helped lay the foundation for the theory of probability. From this brief but productive collaboration on the problem of points, they are now regarded as joint founders of probability theory. Fermat is credited with carrying out the first-ever rigorous probability calculation. In it, he was asked by a professional gambler why if he bet on rolling at least one six in four throws of a die he won in the long term, whereas betting on throwing at least one double-six in 24 throws of two dice resulted in his losing. Fermat showed mathematically why this was the case. The first variational principle in
physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical science is that depar ...
was articulated by
Euclid Euclid (; grc-gre, Wikt:Εὐκλείδης, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the ''Euclid's Elements, Elements'' trea ...
in his ''Catoptrica''. It says that, for the path of light reflecting from a mirror, the angle of incidence equals the angle of reflection. Hero of Alexandria later showed that this path gave the shortest length and the least time. Fermat refined and generalized this to "light travels between two given points along the path of shortest ''time''" now known as the '' principle of least time''. For this, Fermat is recognized as a key figure in the historical development of the fundamental principle of least action in physics. The terms Fermat's principle and ''Fermat functional'' were named in recognition of this role.

## Death

Pierre de Fermat died on January 12, 1665, at Castres, in the present-day department of Tarn.Klaus Barner (2001): ''How old did Fermat become?''
Internationale Zeitschrift für Geschichte und Ethik der Naturwissenschaften, Technik und Medizin. . Vol 9, No 4, pp. 209-228.
The oldest and most prestigious high school in
Toulouse Toulouse ( , ; oc, Tolosa ) is the Prefectures in France, prefecture of the Departments of France, French department of Haute-Garonne and of the larger Regions of France, region of Occitania (administrative region), Occitania. The city is on t ...
is named after him: the Lycée Pierre-de-Fermat. French sculptor Théophile Barrau made a marble statue named ''Hommage à Pierre Fermat'' as a tribute to Fermat, now at the Capitole de Toulouse. File:Fermat burial plaque.jpg, alt=Plaque at the place of burial of Pierre de Fermat , Place of burial of Pierre de Fermat in Place Jean Jaurés, Castres. Translation of the plaque: in this place was buried on January 13, 1665, Pierre de Fermat, councillor at the Chambre de l'Édit (a court established by the Edict of Nantes) and mathematician of great renown, celebrated for his theorem,
an + bn ≠ cn for n>2 File:Beaumont-de-Lomagne - Monument à Fermat.jpg, Monument to Fermat in Beaumont-de-Lomagne in Tarn-et-Garonne, southern France File:Capitole Toulouse - Salle Henri-Martin - Buste de Pierre de Fermat.jpg, Bust in the Salle Henri-Martin in the Capitole de Toulouse File:Fermats will.jpg, Holographic will handwritten by Fermat on 4 March 1660, now kept at the Departmental Archives of Haute-Garonne, in
Toulouse Toulouse ( , ; oc, Tolosa ) is the Prefectures in France, prefecture of the Departments of France, French department of Haute-Garonne and of the larger Regions of France, region of Occitania (administrative region), Occitania. The city is on t ...

# Assessment of his work

Together with
René Descartes René Descartes ( or ; ; Latinisation of names, Latinized: Renatus Cartesius; 31 March 1596 – 11 February 1650) was a French people, French philosopher, scientist, and mathematician, widely considered a seminal figure in the emergence of m ...
, Fermat was one of the two leading mathematicians of the first half of the 17th century. According to Peter L. Bernstein, in his 1996 book ''Against the Gods'', Fermat "was a mathematician of rare power. He was an independent inventor of
analytic geometry In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry using a coordinate system. This contrasts with synthetic geometry. Analytic geometry is used in physics and engineerin ...
, he contributed to the early development of calculus, he did research on the weight of the earth, and he worked on light refraction and optics. In the course of what turned out to be an extended correspondence with
Blaise Pascal Blaise Pascal ( , , ; ; 19 June 1623 – 19 August 1662) was a French mathematician, physicist, inventor, philosopher, and Catholic Church, Catholic writer. He was a child prodigy who was educated by his father, a tax collector in Rouen. Pa ...
, he made a significant contribution to the theory of probability. But Fermat's crowning achievement was in the theory of numbers." Regarding Fermat's work in analysis,
Isaac Newton Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician, physicist, astronomer, alchemist, Theology, theologian, and author (described in his time as a "natural philosophy, natural philosopher"), widely ...
wrote that his own early ideas about calculus came directly from "Fermat's way of drawing tangents." Of Fermat's number theoretic work, the 20th-century mathematician André Weil wrote that: "what we possess of his methods for dealing with curves of genus 1 is remarkably coherent; it is still the foundation for the modern theory of such curves. It naturally falls into two parts; the first one ... may conveniently be termed a method of ascent, in contrast with the descent which is rightly regarded as Fermat's own." Regarding Fermat's use of ascent, Weil continued: "The novelty consisted in the vastly extended use which Fermat made of it, giving him at least a partial equivalent of what we would obtain by the systematic use of the group theoretical properties of the rational points on a standard cubic."Weil 1984, p.105 With his gift for number relations and his ability to find proofs for many of his theorems, Fermat essentially created the modern theory of numbers.

* Diagonal form * Euler's theorem * List of things named after Pierre de Fermat

# References

## Works cited

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Fermat's Achievements

at MathPages
The Correspondence of Pierre de Fermat
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EMLO

History of Fermat's Last Theorem (French)
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from W. W. Rouse Ball's History of Mathematics * {{DEFAULTSORT:Fermat, Pierre 1607 births 1665 deaths 17th-century French mathematicians 17th-century French judges French Roman Catholics History of calculus Number theorists French geometers Occitan people People from Tarn-et-Garonne