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, structure, space, models, and change.
History
On ...
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 Adequality is a technique developed by Pierre de Fermat in his treatise ''Methodus ad disquirendam maximam et minimam''[ordinate
In common usage, the abscissa refers to the (''x'') coordinate and the ordinate refers to the (''y'') coordinate of a standard two-dimensional graph.
The distance of a point from the y-axis, scaled with the x-axis, is called abscissa or x c ...](_blank)
s 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
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 ...
. 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
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 ...
, 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 instruments that use or detect it. Optics usually describes the behaviour of visible, ultraviole ...
. 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 pat ...
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 integers , , and satisfy the equation for any integer value of greater than 2. The cases and have been k ...
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 ...
, 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''. 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
Toulouse ( , ; oc, Tolosa ) is the prefecture of the French department of Haute-Garonne and of the larger region of Occitania. The city is on the banks of the River Garonne, from the Mediterranean Sea, from the Atlantic Ocean and from Par ...
,
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
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
Bordeaux ( , ; Gascon oc, Bordèu ; eu, Bordele; it, Bordò; es, Burdeos) is a port city on the river Garonne in the Gironde department, Southwestern France. It is the capital of the Nouvelle-Aquitaine region, as well as the prefectur ...
. 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
In mathematical analysis, the maxima and minima (the respective plurals of maximum and minimum) of a function, known collectively as extrema (the plural of extremum), are the largest and smallest value of the function, either within a given ran ...
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
François Viète, Seigneur de la Bigotière ( la, Franciscus Vieta; 1540 – 23 February 1603), commonly know by his mononym, Vieta, was a French mathematician whose work on new algebra was an important step towards modern algebra, due to i ...
.
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,
Latin
Latin (, or , ) is a classical language belonging to the 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 the power of the ...
,
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
Italian(s) may refer to:
* Anything of, from, or related to the people of Italy over the centuries
** Italians, an ethnic group or simply a citizen of the Italian Republic or Italian Kingdom
** Italian language, a Romance language
*** Regional Ita ...
and
Spanish
Spanish might refer to:
* Items from or related to Spain:
**Spaniards are a nation and ethnic group indigenous to Spain
**Spanish language, spoken in Spain and many Latin American countries
**Spanish cuisine
Other places
* Spanish, Ontario, Can ...
), 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
Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of ma ...
. 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 engineer ...
, 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
Anders Hjorth Hald (3 July 1913 – 11 November 2007) was a Danish statistician. He was a professor at the University of Copenhagen from 1960 to 1982. While a professor, he did research in industrial quality control and other areas, and also autho ...
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
![Diophantus-II-8-Fermat](https://upload.wikimedia.org/wikipedia/commons/4/47/Diophantus-II-8-Fermat.jpg)
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 Adequality is a technique developed by Pierre de Fermat in his treatise ''Methodus ad disquirendam maximam et minimam''[tangent
In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. More ...](_blank)
s 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
In mathematics, a geometric series is the sum of an infinite number of terms that have a constant ratio between successive terms. For example, the series
:\frac \,+\, \frac \,+\, \frac \,+\, \frac \,+\, \cdots
is geometric, because each succ ...
.
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
Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of ma ...
, when they independently developed the
fundamental theorem of calculus
The fundamental theorem of calculus is a theorem that links the concept of differentiating a function (calculating its slopes, or rate of change at each time) with the concept of integrating a function (calculating the area under its graph, or ...
.
In number theory, Fermat studied
Pell's equation
Pell's equation, also called the Pell–Fermat equation, is any Diophantine equation of the form x^2 - ny^2 = 1, where ''n'' is a given positive nonsquare integer, and integer solutions are sought for ''x'' and ''y''. In Cartesian coordinate ...
,
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 number
Amicable numbers are two different natural numbers related in such a way that the sum of the proper divisors of each is equal to the other number. That is, σ(''a'')=''b'' and σ(''b'')=''a'', where σ(''n'') is equal to the sum of positive d ...
s and what would later become
Fermat numbers
In mathematics, a Fermat number, named after Pierre de Fermat, who first studied them, is a positive integer of the form
:F_ = 2^ + 1,
where ''n'' is a non-negative integer. The first few Fermat numbers are:
: 3, 5, 17, 257, 65537, 42949672 ...
. 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
Fermat's factorization method, named after Pierre de Fermat, is based on the representation of an odd integer as the difference of two squares:
:N = a^2 - b^2.
That difference is algebraically factorable as (a+b)(a-b); if neither factor equals one, ...
—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 f ...
, which he used to prove
Fermat's right triangle theorem
Fermat's right triangle theorem is a non-existence proof in number theory, published in 1670 among the works of Pierre de Fermat, soon after his death. It is the only complete proof given by Fermat. It has several equivalent formulations, one o ...
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 number
A triangular number or triangle number counts objects arranged in an equilateral triangle. Triangular numbers are a type of figurate number, other examples being square numbers and cube numbers. The th triangular number is the number of dots in ...
s,
four square numbers, five
pentagonal number
A pentagonal number is a figurate number that extends the concept of triangular and square numbers to the pentagon, but, unlike the first two, the patterns involved in the construction of pentagonal numbers are not rotationally symmetrical. The ...
s, and so on.
Although Fermat claimed to have proven all his arithmetic theorems, few records of his proofs have survived. Many mathematicians, including
Gauss
Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...
, 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
The problem of points, also called the problem of division of the stakes, is a classical problem in probability theory. One of the famous problems that motivated the beginnings of modern probability theory in the 17th century, it led Blaise Pascal ...
, they are now regarded as joint founders of
probability theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set o ...
.
Fermat is credited with carrying out the first-ever rigorous probability calculation. In it, he was asked by a professional
gambler
Gambling (also known as betting or gaming) is the wagering of something of value ("the stakes") on a random event with the intent of winning something else of value, where instances of strategy are discounted. Gambling thus requires three elem ...
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 science and especially in mathematical studies, a variational principle is one that enables a problem to be solved using calculus of variations, which concerns finding functions that optimize the values of quantities that depend on those func ...
in
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
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
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
Hero of Alexandria (; grc-gre, Ἥρων ὁ Ἀλεξανδρεύς, ''Heron ho Alexandreus'', also known as Heron of Alexandria ; 60 AD) was a Greece, Greek mathematician and engineer who was active in his native city of Alexandria, Roman Egy ...
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
The stationary-action principle – also known as the principle of least action – is a variational principle that, when applied to the '' action'' of a mechanical system, yields the equations of motion for that system. The principle states tha ...
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 pat ...
and ''Fermat functional'' were named in recognition of this role.
Death
Pierre de Fermat died on January 12, 1665, at
Castres
Castres (; ''Castras'' in the Languedocian dialect, Languedocian dialect of Occitan language, Occitan) is the sole Subprefectures in France, subprefecture of the Tarn (department), Tarn Departments of France, department in the Occitania (administ ...
, 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 prefecture of the French department of Haute-Garonne and of the larger region of Occitania. The city is on the banks of the River Garonne, from the Mediterranean Sea, from the Atlantic Ocean and from Par ...
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 i ...
. 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
300px, The Capitole back side
The Capitole de Toulouse ( oc, Capitòli de Tolosa, link=no; ), commonly known as the ''Capitole'', is the heart of the municipal administration of the French city of Toulouse and its city hall.
__NOTOC__
His ...
.
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
Castres (; ''Castras'' in the Languedocian dialect, Languedocian dialect of Occitan language, Occitan) is the sole Subprefectures in France, subprefecture of the Tarn (department), Tarn Departments of France, department in the Occitania (administ ...
. 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
The Edict of Nantes () was signed in April 1598 by King Henry IV and granted the Calvinist Protestants of France, also known as Huguenots, substantial rights in the nation, which was in essence completely Catholic. In the edict, Henry aimed pr ...
) 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
Tarn-et-Garonne (; oc, Tarn e Garona ) is a department in the Occitania region in Southern France. It is traversed by the rivers Tarn and Garonne, from which it takes its name. The area was originally part of the former provinces of Quercy and ...
, 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
300px, The Capitole back side
The Capitole de Toulouse ( oc, Capitòli de Tolosa, link=no; ), commonly known as the ''Capitole'', is the heart of the municipal administration of the French city of Toulouse and its city hall.
__NOTOC__
His ...
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
Haute-Garonne (; oc, Nauta Garona, ; en, Upper Garonne) is a department in the Occitanie region of Southwestern France. Named after the river Garonne, which flows through the department. Its prefecture and main city is Toulouse, the country's ...
, in Toulouse
Toulouse ( , ; oc, Tolosa ) is the prefecture of the French department of Haute-Garonne and of the larger region of Occitania. The city is on the banks of the River Garonne, from the Mediterranean Sea, from the Atlantic Ocean and from Par ...
Assessment of his work
Together with
René Descartes
René Descartes ( or ; ; Latinized: Renatus Cartesius; 31 March 1596 – 11 February 1650) was a French philosopher, scientist, and mathematician, widely considered a seminal figure in the emergence of modern philosophy and science. Mathem ...
, 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, theologian, and author (described in his time as a "natural philosopher"), widely recognised as one of the grea ...
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 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
**Heritag ...
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 point
In number theory and algebraic geometry, a rational point of an algebraic variety is a point whose coordinates belong to a given field. If the field is not mentioned, the field of rational numbers is generally understood. If the field is the field ...
s 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
In mathematics, a diagonal form is an algebraic form (homogeneous polynomial) without cross-terms involving different indeterminates. That is, it is
:\sum_^n a_i ^m\
for some given degree ''m''.
Such forms ''F'', and the hypersurfaces ''F'' = ...
*
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
This is a list of things named after Pierre de Fermat, a French amateur mathematician.
*Director circle, Fermat–Apollonius circle
*Fermat–Catalan conjecture
*Fermat cubic
*Fermat curve
*Euler's theorem, Fermat–Euler theorem
*Fermat number
*F ...
Notes
References
Works cited
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Further reading
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External links
Fermat's Achievementsat MathPages
The Correspondence of Pierre de Fermat i
EMLOHistory of Fermat's Last Theorem (French)* Th
from
W. W. Rouse Ball's History of Mathematics
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{{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