Scientists



FERMAT
Pierre de Fermat (1601 or 1607/8 – 1665) was a French lawyer at the Parlement of Toulouse, France, and an amateur mathematician who is
given credit for early developments that led to infinitesimal calculus, including his 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 the then
unknown differential calculus, and his research into number theory. He made notable contributions to analytic geometry, probability, and
optics. He is best known for Fermat's Last Theorem, which he described in a note at the margin of a copy of Diophantus' Arithmetic.
Fermat was born in BeaumontdeLomagne, TarnetGaronne, France; the late 15th century mansion where Fermat was born is now a museum. He was
of Basque origin Fermat's father was a wealthy leather merchant and second consul of BeaumontdeLomagne. Pierre had a brother and two sisters
and was almost certainly brought up in the town of his birth. There is little evidence concerning his school education, but it may have been
at the local Franciscan monastery.
He attended the University of Toulouse before moving to Bordeaux in the second half of the 1620s. 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
Etienne d'Espagnet who clearly shared mathematical interests with Fermat. There he became much influenced by the work of Francois Viete.
From Bordeaux, Fermat went to Orleans where he studied law at the University. He received a degree in civil law before, in 1631, receiving
the title of councillor at the High Court of Judicature in Toulouse, which he held for the rest of his life. Due to the office he now held he
became entitled to change his name from Pierre Fermat to Pierre de Fermat. Fluent in Latin, Basque, 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. This allowed him to preserve his status
as an "amateur" while gaining the recognition he desired. This naturally led to priority disputes with contemporaries such as Descartes and
Wallis. He developed a close relationship with Blaise Pascal.
Anders Hald writes that, "The basis of Fermat's mathematics was the classical Greek treatises combined with Vieta's new algebraic methods".
Fermat's pioneering work in analytic geometry was circulated in manuscript form in 1636, predating the publication of Descartes' famous La
geometrie. 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 minima and in De tangentibus linearum curvarum, Fermat developed a method for determining maxima,
minima, and tangents to various curves that was equivalent to differentiation. 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. Using an ingenious trick, 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 numbers, amicable numbers and what would later become Fermat numbers. It was while researching perfect numbers that he discovered the
little theorem. He invented a factorization method – Fermat's factorization method – as well as the proof technique of infinite descent, which
he used to prove Fermat's Last Theorem for the case n = 4. Fermat developed the twosquare theorem, and the polygonal number theorem,
which states that each number is a sum of three triangular numbers, four square numbers, fivepentagonal numbers, and so on.
Although Fermat claimed to have proved 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 tools available to
Fermat. His famous Last Theorem was first discovered by his son in the margin on his father's copy of an edition of Diophantus, and included
the statement that the margin was too small to include the proof. He had not bothered to inform even Marin Mersenne of it. It was not proved
until 1994, using techniques unavailable to Fermat. His Last Theorem states: "Diophantus' equation _{}
does not have natural decisions x, y and z at n > 2". Although he carefully studied, and drew inspiration from
Diophantus, Fermat began a different tradition. Diophantus was content to find a single solution to his equations, even if it were an
undesired fractional one. Fermat was interested only in integer solutions to his Diophantine equations, and he looked for all possible general
solutions. He often proved that certain equations had no solution, which usually baffled his contemporaries.
Through his correspondence with Pascal in 1654, Fermat and Pascal helped lay the fundamental groundwork 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 doublesix in 24 throws of
two dice resulted in his losing. Fermat subsequently proved why this was the case mathematically.
Fermat's principle of least time (which he used to derive Snell's law in 1657) was the first variational principle enunciated in physics
since Hero of Alexandria described a principle of least distance in the first century CE. In this way, 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.
He died at Castres, Tarn. The oldest and most prestigious high school in Toulouse is named after him: the Lycee Pierre de Fermat. French
sculptor Theophile Barrau made a marble statue named Hommage a Pierre Fermat as tribute to Fermat, now at the Capitole of Toulouse.
Together with Rene Descartes, Fermat was one of the two leading mathematicians of the first half of the 17th century. According to Peter L.
Bernstein, in his book Against the Gods, Fermat "was a mathematician of rare power. He was an independent inventor of analytic geometry, 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 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 great 20thcentury mathematician Andre 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". 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.
