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Pierre de Fermat (; between 31 October and 6 December 1607 – 12 January 1665) was a
French French (french: français(e), link=no) may refer to: * Something of, from, or related to France ** French language, which originated in France, and its various dialects and accents ** French people, a nation and ethnic group identified with Franc ...
mathematician who is given credit for early developments that led to
infinitesimal calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithm ...
, 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 In mathematics, differential calculus is a subfield of calculus that studies the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral calculus—the study of the area beneath a curve. ...
, then unknown, and his research into number theory. 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. He is best known for his Fermat's principle for light propagation and his Fermat's Last Theorem in number theory, which he described in a note at the margin of a copy of
Diophantus Diophantus of Alexandria ( grc, Διόφαντος ὁ Ἀλεξανδρεύς; born probably sometime between AD 200 and 214; died around the age of 84, probably sometime between AD 284 and 298) was an Alexandrian mathematician, who was the aut ...
' ''
Arithmetica ''Arithmetica'' ( grc-gre, Ἀριθμητικά) is an Ancient Greek text on mathematics written by the mathematician Diophantus () in the 3rd century AD. It is a collection of 130 algebraic problems giving numerical solutions of determinate e ...
''. He was also a lawyer at the ''
Parlement A ''parlement'' (), under the French Ancien Régime, was a provincial appellate court of the Kingdom of France. In 1789, France had 13 parlements, the oldest and most important of which was the Parlement of Paris. While both the modern Fre ...
'' of Toulouse, France.


Biography

Fermat was born in 1607 in
Beaumont-de-Lomagne Beaumont-de-Lomagne (; Languedocien: ''Bèumont de Lomanha'') is a commune in the Tarn-et-Garonne department in the Occitanie region in southern France. Geography The river Gimone runs through the town. History Beaumont-de-Lomagne, bastide, was ...
, France—the late 15th-century mansion where Fermat was born is now a museum. He was from
Gascony Gascony (; french: Gascogne ; oc, Gasconha ; eu, Gaskoinia) was a province of the southwestern Kingdom of France that succeeded the Duchy of Gascony (602–1453). From the 17th century until the French Revolution (1789–1799), it was part o ...
, 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 The University of Orléans (french: Université d'Orléans) is a French university, in the Academy of Orléans and Tours. As of July 2015 it is a member of the regional university association Leonardo da Vinci consolidated University. History ...
from 1623 and received a bachelor in civil law in 1626, before moving to Bordeaux. 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 Étienne d'Espagnet (born c. 1596) was the son of parliamentary counselor Jean d'Espagnet and Charlotte De Mangeau. He became a parliamentary counselor in 1617. He married in 1629 and had a son in 1634. He was friends with Viète and Fermat P ...
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 A councillor is an elected representative for a local government council in some countries. Canada Due to the control that the provinces have over their municipal governments, terms that councillors serve vary from province to province. Unl ...
at the
Parlement de Toulouse The Parliament of Toulouse (french: Parlement de Toulouse) was one of the ''parlements'' of the Kingdom of France, established in the city of Toulouse. It was modelled on the Parliament of Paris. It was first created in 1420, but definitely estab ...
, 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 French (french: français(e), link=no) may refer to: * Something of, from, or related to France ** French language, which originated in France, and its various dialects and accents ** French people, a nation and ethnic group identified with Franc ...
, Latin,
Occitan Occitan may refer to: * Something of, from, or related to the Occitania territory in parts of France, Italy, Monaco and Spain. * Something of, from, or related to the Occitania administrative region of France. * Occitan language Occitan (; o ...
, 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 Newton most commonly refers to: * Isaac Newton (1642–1726/1727), English scientist * Newton (unit), SI unit of force named after Isaac Newton Newton may also refer to: Arts and entertainment * ''Newton'' (film), a 2017 Indian film * 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 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, 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 algebra New is an adjective referring to something recently made, discovered, or created. New or NEW may refer to: Music * New, singer of K-pop group The Boyz Albums and EPs * ''New'' (album), by Paul McCartney, 2013 * ''New'' (EP), by Regurgitator, ...
ic 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 ''La Géométrie'' was published in 1637 as an appendix to ''Discours de la méthode'' (''Discourse on the Method''), written by René Descartes. In the ''Discourse'', he presents his method for obtaining clarity on any subject. ''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 tangents to various curves that was equivalent to
differential calculus In mathematics, differential calculus is a subfield of calculus that studies the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral calculus—the study of the area beneath a curve. ...
. 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 Newton most commonly refers to: * Isaac Newton (1642–1726/1727), English scientist * Newton (unit), SI unit of force named after Isaac Newton Newton may also refer to: Arts and entertainment * ''Newton'' (film), a 2017 Indian film * Newton ( ...
, and then Leibniz, when they independently developed the fundamental theorem of calculus. In number theory, Fermat studied Pell's equation, perfect numbers, amicable numbers and what would later become Fermat numbers. It was while researching perfect numbers that he discovered
Fermat's little theorem Fermat's little theorem states that if ''p'' is a prime number, then for any integer ''a'', the number a^p - a is an integer multiple of ''p''. In the notation of modular arithmetic, this is expressed as : a^p \equiv a \pmod p. For example, if = ...
. He invented a factorization method— Fermat's factorization method—and popularized the proof by
infinite descent In mathematics, a proof by infinite descent, also known as Fermat's method of descent, is a particular kind of proof by contradiction used to show that a statement cannot possibly hold for any number, by showing that if the statement were to hold fo ...
, 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 In additive number theory, the Fermat polygonal number theorem states that every positive integer is a sum of at most Polygonal number, -gonal numbers. That is, every positive integer can be written as the sum of three or fewer triangular number ...
, 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 Diophantus of Alexandria ( grc, Διόφαντος ὁ Ἀλεξανδρεύς; born probably sometime between AD 200 and 214; died around the age of 84, probably sometime between AD 284 and 298) was an Alexandrian mathematician, who was the aut ...
, and included the statement that the margin was too small to include the proof. It seems that he had not written to
Marin Mersenne Marin Mersenne, OM (also known as Marinus Mersennus or ''le Père'' Mersenne; ; 8 September 1588 – 1 September 1648) was a French polymath whose works touched a wide variety of fields. He is perhaps best known today among mathematicians for ...
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 Dice (singular die or dice) are small, throwable objects with marked sides that can rest in multiple positions. They are used for generating random values, commonly as part of tabletop games, including dice games, board games, role-playing g ...
resulted in his losing. Fermat showed mathematically why this was the case. The first variational principle in physics was articulated by Euclid in his ''Catoptrica''. It says that, for the path of light reflecting from a mirror, the
angle of incidence Angle of incidence is a measure of deviation of something from "straight on" and may refer to: * Angle of incidence (aerodynamics), angle between a wing chord and the longitudinal axis, as distinct from angle of attack In fluid dynamics, ang ...
equals the
angle of reflection Reflection is the change in direction of a wavefront at an interface between two different media so that the wavefront returns into the medium from which it originated. Common examples include the reflection of light, sound and water waves. The ' ...
. 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 Fermat's principle, also known as the principle of least time, is the link between ray optics and wave optics. In its original "strong" form, Fermat's principle states that the path taken by a ray between two given points is the pat ...
''. 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 is named after him: the
Lycée Pierre-de-Fermat The Lycée Pierre-de-Fermat, also referred to simply as Pierre-de-Fermat, is a public Lycée, located in the ''Parvis des Jacobins'' in Toulouse, in the immediate vicinity of the Place du Capitole; It occupies a large space in the city center inc ...
. French sculptor
Théophile Barrau Théophile Barrau (1848–1913) was a French sculptor. Barrau was born in Carcassonne. He was a student of Alexandre Falguière and started at the Salon in 1874. He received awards in 1879, 1880, 1889, and became a Chevalier of the Legion o ...
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 Beaumont-de-Lomagne (; Languedocien: ''Bèumont de Lomanha'') is a commune in the Tarn-et-Garonne department in the Occitanie region in southern France. Geography The river Gimone runs through the town. History Beaumont-de-Lomagne, bastide, was ...
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 A holographic will, or olographic testament, is a will and testament which is a holographic document, i.e. it has been entirely handwritten and signed by the testator. Historically, a will had to be signed by witnesses attesting to the validity ...
handwritten by Fermat on 4 March 1660, now kept at the Departmental Archives of Haute-Garonne, in Toulouse


Assessment of his work

Together with René Descartes, Fermat was one of the two leading mathematicians of the first half of the 17th century. According to
Peter L. Bernstein Peter Lewyn Bernstein (January 22, 1919 – June 5, 2009) was an American financial historian, economist and educator whose development and refinement of the efficient-market hypothesis made him one of the country's best known authorities in p ...
, 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 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 André Weil (; ; 6 May 1906 – 6 August 1998) was a French mathematician, known for his foundational work in number theory and algebraic geometry. He was a founding member and the ''de facto'' early leader of the mathematical Bourbaki group. Th ...
wrote that: "what we possess of his methods for dealing with
curves A curve is a geometrical object in mathematics. Curve(s) may also refer to: Arts, entertainment, and media Music * Curve (band), an English alternative rock music group * ''Curve'' (album), a 2012 album by Our Lady Peace * "Curve" (song), a 20 ...
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 Descent may refer to: As a noun Genealogy and inheritance * Common descent, concept in evolutionary biology * Kinship, one of the major concepts of cultural anthropology **Pedigree chart or family tree **Ancestry **Lineal descendant **Heritage (d ...
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.


See also

* Diagonal form *
Euler's theorem In number theory, Euler's theorem (also known as the Fermat–Euler theorem or Euler's totient theorem) states that, if and are coprime positive integers, and \varphi(n) is Euler's totient function, then raised to the power \varphi(n) is congru ...
* List of things named after Pierre de Fermat


Notes


References


Works cited

*


Further reading

* * *


External links


Fermat's Achievements


at MathPages
The Correspondence of Pierre de Fermat
i
EMLO

History of Fermat's Last Theorem (French)
* Th

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