Pierre de Fermat

born August 17, 1601, Beaumont-de-Lomagne, France

died January 12, 1665, Castres

French mathematician who is often called the founder of the
modern theory of numbers. Together with René Descartes, Fermat
was one of the two leading mathematicians of the first half of
the 17th century. Independently of Descartes, Fermat discovered
the fundamental principle of analytic geometry. His methods for
finding tangents to curves and their maximum and minimum points
led him to be regarded as the inventor of the differential
calculus. Through his correspondence with Blaise Pascal he was a
co-founder of the theory of probability.

Life and early work

Little is known of Fermat’s early life and education. He was of
Basque origin and received his primary education in a local
Franciscan school. He studied law, probably at Toulouse and
perhaps also at Bordeaux. Having developed tastes for foreign
languages, classical literature, and ancient science and
mathematics, Fermat followed the custom of his day in composing
conjectural “restorations” of lost works of antiquity. By 1629
he had begun a reconstruction of the long-lost Plane Loci of
Apollonius, the Greek geometer of the 3rd century bc. He soon
found that the study of loci, or sets of points with certain
characteristics, could be facilitated by the application of
algebra to geometry through a coordinate system. Meanwhile,
Descartes had observed the same basic principle of analytic
geometry, that equations in two variable quantities define plane
curves. Because Fermat’s Introduction to Loci was published
posthumously in 1679, the exploitation of their discovery,
initiated in Descartes’s Géométrie of 1637, has since been known
as Cartesian geometry.

In 1631 Fermat received the baccalaureate in law from the
University of Orléans. He served in the local parliament at
Toulouse, becoming councillor in 1634. Sometime before 1638 he
became known as Pierre de Fermat, though the authority for this
designation is uncertain. In 1638 he was named to the Criminal
Court.

Analyses of curves.

Fermat’s study of curves and equations prompted him to
generalize the equation for the ordinary parabola ay = x2, and
that for the rectangular hyperbola xy = a2, to the form an - 1y
= xn. The curves determined by this equation are known as the
parabolas or hyperbolas of Fermat according as n is positive or
negative. He similarly generalized the Archimedean spiral r =
aθ. These curves in turn directed him in the middle 1630s to an
algorithm, or rule of mathematical procedure, that was
equivalent to differentiation. This procedure enabled him to
find equations of tangents to curves and to locate maximum,
minimum, and inflection points of polynomial curves, which are
graphs of linear combinations of powers of the independent
variable. During the same years, he found formulas for areas
bounded by these curves through a summation process that is
equivalent to the formula now used for the same purpose in the
integral calculus. Such a formula is:

It is not known whether or not Fermat noticed that
differentiation of xn, leading to nan - 1, is the inverse of
integrating xn. Through ingenious transformations he handled
problems involving more general algebraic curves, and he applied
his analysis of infinitesimal quantities to a variety of other
problems, including the calculation of centres of gravity and
finding the lengths of curves. Descartes in the Géométrie had
reiterated the widely held view, stemming from Aristotle, that
the precise rectification or determination of the length of
algebraic curves was impossible; but Fermat was one of several
mathematicians who, in the years 1657–59, disproved the dogma.
In a paper entitled “De Linearum Curvarum cum Lineis Rectis
Comparatione” (“Concerning the Comparison of Curved Lines with
Straight Lines”), he showed that the semicubical parabola and
certain other algebraic curves were strictly rectifiable. He
also solved the related problem of finding the surface area of a
segment of a paraboloid of revolution. This paper appeared in a
supplement to the Veterum Geometria Promota, issued by the
mathematician Antoine de La Loubère in 1660. It was Fermat’s
only mathematical work published in his lifetime.

Disagreement with other Cartesian views

Fermat differed also with Cartesian views concerning the law of
refraction (the sines of the angles of incidence and refraction
of light passing through media of different densities are in a
constant ratio), published by Descartes in 1637 in La
Dioptrique; like La Géométrie, it was an appendix to his
celebrated Discours de la méthode. Descartes had sought to
justify the sine law through a premise that light travels more
rapidly in the denser of the two media involved in the
refraction. Twenty years later Fermat noted that this appeared
to be in conflict with the view espoused by Aristotelians that
nature always chooses the shortest path. Applying his method of
maxima and minima and making the assumption that light travels
less rapidly in the denser medium, Fermat showed that the law of
refraction is consonant with his “principle of least time.” His
argument concerning the speed of light was found later to be in
agreement with the wave theory of the 17th-century Dutch
scientist Christiaan Huygens, and in 1849 it was verified
experimentally by A.-H.-L. Fizeau.

Through the mathematician and theologian Marin Mersenne, who,
as a friend of Descartes, often acted as an intermediary with
other scholars, Fermat in 1638 maintained a controversy with
Descartes on the validity of their respective methods for
tangents to curves. Fermat’s views were fully justified some 30
years later in the calculus of Sir Isaac Newton. Recognition of
the significance of Fermat’s work in analysis was tardy, in part
because he adhered to the system of mathematical symbols devised
by François Viète, notations that Descartes’s Géométrie had
rendered largely obsolete. The handicap imposed by the awkward
notations operated less severely in Fermat’s favourite field of
study, the theory of numbers; but here, unfortunately, he found
no correspondent to share his enthusiasm. In 1654 he had enjoyed
an exchange of letters with his fellow mathematician Blaise
Pascal on problems in probability concerning games of chance,
the results of which were extended and published by Huygens in
his De Ratiociniis in Ludo Aleae (1657).

Work on theory of numbers

Fermat vainly sought to persuade Pascal to join him in research
in number theory. Inspired by an edition in 1621 of the
Arithmetic of Diophantus, the Greek mathematician of the 3rd
century ad, Fermat had discovered new results in the so-called
higher arithmetic, many of which concerned properties of prime
numbers (those positive integers that have no factors other than
1 and themselves). One of the most elegant of these had been the
theorem that every prime of the form 4n + 1 is uniquely
expressible as the sum of two squares. A more important result,
now known as Fermat’s lesser theorem, asserts that if p is a
prime number and if a is any positive integer, then ap - a is
divisible by p. Fermat seldom gave demonstrations of his
results, and in this case proofs were provided by Gottfried
Leibniz, the 17th-century German mathematician and philosopher,
and Leonhard Euler, the 18th-century Swiss mathematician. For
occasional demonstrations of his theorems Fermat used a device
that he called his method of “infinite descent,” an inverted
form of reasoning by recurrence or mathematical induction. One
unproved conjecture by Fermat turned out to be false. In 1640,
in letters to mathematicians and to other knowledgeable thinkers
of the day, including Blaise Pascal, he announced his belief
that numbers of the form 22n + 1, known since as “numbers of
Fermat,” are necessarily prime; but a century later Euler showed
that 225 + 1 has 641 as a factor. It is not known if there are
any primes among the Fermat numbers for n > 5. Carl Friedrich
Gauss in 1796 in Germany found an unexpected application for
Fermat numbers when he showed that a regular polygon of N sides
is constructible in a Euclidean sense if N is a prime Fermat
number or a product of distinct Fermat primes. By far the best
known of Fermat’s many theorems is a problem known as his
“great,” or “last,” theorem. This appeared in the margin of his
copy of Diophantus’ Arithmetica and states that the equation xn
+ yn = zn, where x, y, z, and n are positive integers, has no
solution if n is greater than 2. This theorem remained unsolved
until the late 20th century.

Fermat was the most productive mathematician of his day. But
his influence was circumscribed by his reluctance to publish.

**Carl B. Boyer**