Pierre de Fermat
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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 France (), officially the French Republic (french: link=no, République française), is a country primarily located in Western Europe, consistin ...
mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry) ...

mathematician
who is given credit for early developments that led to
infinitesimal calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimal In mathematics, infinitesimals or infinitesimal numbers are quantities that are closer to zero than any standard real number, but are not zero. They do not ex ...
, including his technique of
adequalityAdequality is a technique developed by Pierre de Fermat in his treatise ''Methodus ad disquirendam maximam et minimam''
. In particular, he is recognized for his discovery of an original method of finding the greatest and the smallest
ordinate In common usage, the abscissa refers to the horizontal (''x'') axis and the ordinate refers to the vertical (''y'') axis of a standard two-dimensional graph. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of suc ...
s of curved lines, which is analogous to that of
differential calculus In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
, 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 integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777– ...

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 Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measur ...
,
probability Probability is the branch of mathematics concerning numerical descriptions of how likely an Event (probability theory), event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and ...

probability
, and
optics Optics is the branch of physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its Motion (physics), motion a ...

optics
. He is best known for his
Fermat's principle 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 path ...
for light propagation and his
Fermat's Last Theorem In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive number, positive integers , , and satisfy the equation for any integer value of greater than 2. The cases ...
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 integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777– ...

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 autho ...
' ''
Arithmetica ''Arithmetica'' ( grc-gre, Ἀριθμητικά) is an Ancient Greek Ancient Greek includes the forms of the Greek language used in ancient Greece and the classical antiquity, ancient world from around 1500 BC to 300 BC. It is often roug ...

Arithmetica
''. He was also a lawyer at the ''
Parlement A ''parlement'' (), under the French Ancien Régime File:Prise de la Bastille.jpg, The ''Storming of the Bastille'' on 14 July 1789, later taken to mark the end of the ''Ancien Régime''; watercolour by Jean-Pierre Houël The Ancien Rég ...
'' 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 Occitanie. The city is on the banks of the Garonne, River Garon ...

Toulouse
,
France France (), officially the French Republic (french: link=no, République française), is a country primarily located in Western Europe, consisting of metropolitan France and Overseas France, several overseas regions and territories. The metro ...

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 Tarn-et-Garonne (; oc, Tarn e Garona ) is a department Southwestern France France (), officially the French Republic (french: link=no, R ...
, 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 A province is almost always an administrative division within a country or state. The term derives from the ancient Roman '' provincia'', which was the major terr ...
, 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 oc, Bordèu ) is a port city on the river Garonne The Garonne (, also , ; Occitan Occitan (; oc, occitan, link=no , ), also known as ''lenga d'òc'' (; french: langue d'oc) by its native speakers, is a Romance l ...

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 Jean de Beaugrand, 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 language, French, Latin, Occitan language, Occitan, classical Greek, Italian language, Italian and Spanish language, 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 Isaac Newton, Newton or Gottfried Wilhelm Leibniz, 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 René Descartes, Descartes and John Wallis, 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 Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measur ...
(''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'' (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 (
adequalityAdequality is a technique developed by Pierre de Fermat in his treatise ''Methodus ad disquirendam maximam et minimam''
) for determining maxima, minima, and tangents to various curves that was equivalent to
differential calculus In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
. 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 (mathematics), 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 Isaac Newton, Newton, and then Gottfried Wilhelm Leibniz, 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. 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 Fermat's theorem on sums of two squares, two-square theorem, and the Fermat polygonal number theorem, polygonal number theorem, which states that each number is a sum of three triangular numbers, Lagrange's four-square theorem, 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 Carl Friedrich Gauss, 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 Fermat's Last Theorem, 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 autho ...
, 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 Wiles's proof of Fermat's Last Theorem, Sir Andrew Wiles, using techniques unavailable to Fermat. Through their correspondence in 1654, Fermat and Blaise Pascal 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 History of variational principles in physics, 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 (optics), 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 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 path ...
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 (department), 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 Occitanie. The city is on the banks of the Garonne, River Garon ...

Toulouse
is named after him: the . 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 Beaumont-de-Lomagne (; Languedocien: ''Bèumont de Lomanha'') is a commune in the Tarn-et-Garonne Tarn-et-Garonne (; oc, Tarn e Garona ) is a department Southwestern France France (), officially the French Republic (french: link=no, R ...
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 Occitanie. The city is on the banks of the Garonne, River Garon ...

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, 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 Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measur ...
, 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, 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 wrote that: "what we possess of his methods for dealing with Algebraic curve, curves of Elliptic curve, 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 infinite descent, 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 theory, 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 * 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
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EMLO

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

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