Pierre Fermat's father was a wealthy leather merchant
and second consul of Beaumont- de- Lomagne. Pierre had a brother and two
sisters and was almost certainly brought up in the town of his birth. Although
there is little evidence concerning his school education it must 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 Plane loci 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
who clearly shared mathematical interests with Fermat.

From Bordeaux Fermat went to Orléans where he
studied law at the University. He received a degree in civil law and he
purchased the offices of councillor at the parliament in Toulouse. So by 1631
Fermat was a lawyer and government official in Toulouse and because of the
office he now held he became entitled to change his name from Pierre Fermat to
Pierre de Fermat.

For the remainder of his life he lived in Toulouse but
as well as working there he also worked in his home town of Beaumont-de-Lomagne
and a nearby town of Castres. From his appointment on 14 May 1631 Fermat worked
in the lower chamber of the parliament but on 16 January 1638 he was appointed
to a higher chamber, then in 1652 he was promoted to the highest level at the
criminal court. Still further promotions seem to indicate a fairly meteoric
rise through the profession but promotion was done mostly on seniority and the
plague struck the region in the early 1650s meaning that many of the older men
died. Fermat himself was struck down by the plague and in 1653 his death was
wrongly reported, then corrected:-

I informed you earlier of the death of Fermat. He is
alive, and we no longer fear for his health, even though we had counted him
among the dead a short time ago.

The following report, made to Colbert the leading
figure in France at the time, has a ring of truth:-

Fermat, a man of great erudition, has contact with men
of learning everywhere. But he is rather preoccupied, he does not report cases
well and is confused.

Of course Fermat was preoccupied with mathematics. He
kept his mathematical friendship with
Beaugrand after he moved to Toulouse but there he gained a new
mathematical friend in Carcavi. Fermat
met Carcavi in a professional capacity
since both were councillors in Toulouse but they both shared a love of
mathematics and Fermat told Carcavi
about his mathematical discoveries.

In 1636
Carcavi went to Paris as royal librarian and made contact with Mersenne and his group. Mersenne's interest was aroused by Carcavi's descriptions of Fermat's
discoveries on falling bodies, and he wrote to Fermat. Fermat replied on 26
April 1636 and, in addition to telling
Mersenne about errors which he believed that Galileo had made in his description of free fall, he also
told Mersenne about his work on spirals
and his restoration of Apollonius's
Plane loci. His work on spirals had been motivated by considering the path of
free falling bodies and he had used methods generalised from Archimedes' work On spirals to compute areas
under the spirals. In addition Fermat wrote:-

I have also found many sorts of analyses for diverse
problems, numerical as well as geometrical, for the solution of which Viète's analysis could not have
sufficed. I will share all of this with you whenever you wish and do so without
any ambition, from which I am more exempt and more distant than any man in the
world.

It is somewhat ironical that this initial contact with
Fermat and the scientific community came through his study of free fall since
Fermat had little interest in physical applications of mathematics. Even with
his results on free fall he was much more interested in proving geometrical
theorems than in their relation to the real world. This first letter did
however contain two problems on maxima which Fermat asked Mersenne to pass on to the Paris
mathematicians and this was to be the typical style of Fermat's letters, he
would challenge others to find results which he had already obtained.

Roberval
and Mersenne found that Fermat's
problems in this first, and subsequent, letters were extremely difficult and
usually not soluble using current techniques. They asked him to divulge his
methods and Fermat sent Method for determining Maxima and Minima and Tangents to Curved Lines, his restored text
of Apollonius's Plane loci and his
algebraic approach to geometry Introduction to Plane and Solid Loci to the
Paris mathematicians.

His reputation as one of the leading mathematicians in
the world came quickly but attempts to get his work published failed mainly
because Fermat never really wanted to put his work into a polished form.
However some of his methods were published, for example Hérigone added a supplement
containing Fermat's methods of maxima and minima to his major work Cursus
mathematicus. The widening correspondence between Fermat and other
mathematicians did not find universal praise.
Frenicle de Bessy became annoyed at Fermat's problems which to him were
impossible. He wrote angrily to Fermat but although Fermat gave more details in
his reply, Frenicle de Bessy felt that
Fermat was almost teasing him.

However Fermat soon became engaged in a controversy
with a more major mathematician than
Frenicle de Bessy. Having been sent a copy of Descartes' La Dioptrique by
Beaugrand, Fermat paid it little attention since he was in the middle of
a correspondence with Roberval and Étienne Pascal over methods of
integration and using them to find centres of gravity. Mersenne asked him to give an opinion on La
Dioptrique which Fermat did describing it as
groping about in the shadows.

He claimed that
Descartes had not correctly deduced his law of refraction since it was inherent in his assumptions. To say
that Descartes was not pleased is an
understatement. Descartes soon found
reason to feel even more angry since he viewed Fermat's work on maxima, minima
and tangents as reducing the importance of his own work La
Géométrie which Descartes
was most proud of and which he sought to show that his Discours de la
méthod alone could give.

Descartes
attacked Fermat's method of maxima, minima and tangents. Roberval and Étienne Pascal became involved in the argument and
eventually so did Desargues who Descartes asked to act as a referee. Fermat
proved correct and eventually Descartes
admitted this writing:-

... seeing the last method that you use for finding
tangents to curved lines, I can reply to it in no other way than to say that it
is very good and that, if you had explained it in this manner at the outset, I
would have not contradicted it at all.

Did this end the matter and increase Fermat's
standing? Not at all since Descartes
tried to damage Fermat's reputation. For example, although he wrote to Fermat
praising his work on determining the tangent to a cycloid (which is indeed
correct), Descartes wrote to Mersenne claiming that it was incorrect and
saying that Fermat was inadequate as a mathematician and a thinker. Descartes was important and respected and
thus was able to severely damage Fermat's reputation.

The period from 1643 to 1654 was one when Fermat was
out of touch with his scientific colleagues in Paris. There are a number of
reasons for this. Firstly pressure of work kept him from devoting so much time
to mathematics. Secondly the Fronde, a civil war in France, took place and from
1648 Toulouse was greatly affected. Finally there was the plague of 1651 which
must have had great consequences both on life in Toulouse and of course its
near fatal consequences on Fermat himself. However it was during this time that
Fermat worked on number theory.

Fermat is best remembered for this work in number
theory, in particular for Fermat's Last
Theorem. This theorem states that

xn + yn = zn

has no non-zero integer solutions for x, y and z when
n > 2. Fermat wrote, in the margin of
Bachet's translation of
Diophantus's Arithmetica

I have discovered a truly remarkable proof which this
margin is too small to contain.

These marginal notes only became known after Fermat's
son Samuel published an edition of
Bachet's translation of
Diophantus's Arithmetica with his father's notes in 1670.

It is now believed that Fermat's 'proof' was wrong
although it is impossible to be completely certain. The truth of Fermat's
assertion was proved in June 1993 by the British mathematician Andrew Wiles, but Wiles withdrew the claim to have a proof when problems emerged
later in 1993. In November 1994 Wiles
again claimed to have a correct proof which has now been accepted.

Unsuccessful attempts to prove the theorem over a 300
year period led to the discovery of commutative ring theory and a wealth of
other mathematical discoveries.

Fermat's correspondence with the Paris mathematicians
restarted in 1654 when Blaise Pascal, Étienne Pascal's son, wrote to him to
ask for confirmation about his ideas on
probability. Blaise Pascal knew
of Fermat through his father, who had died three years before, and was well
aware of Fermat's outstanding mathematical abilities. Their short correspondence
set up the theory of probability and from this they are now regarded as joint
founders of the subject. Fermat however, feeling his isolation and still
wanting to adopt his old style of challenging mathematicians, tried to change
the topic from probability to number theory.
Pascal was not interested but Fermat, not realising this, wrote to Carcavi saying:-

I am delighted to have had opinions conforming to
those of M Pascal, for I have infinite
esteem for his genius... the two of you may undertake that publication, of
which I consent to your being the masters, you may clarify or supplement
whatever seems too concise and relieve me of a burden that my duties prevent me
from taking on.

However Pascal
was certainly not going to edit Fermat's work and after this flash of desire to
have his work published Fermat again gave up the idea. He went further than
ever with his challenge problems however:-

Two mathematical problems posed as insoluble to
French, English, Dutch and all mathematicians of Europe by Monsieur de Fermat,
Councillor of the King in the Parliament of Toulouse.

His problems did not prompt too much interest as most
mathematicians seemed to think that number theory was not an important topic.
The second of the two problems, namely to find all solutions of Nx2
+ 1 = y2 for N not a square, was however solved by Wallis and
Brouncker and they developed
continued fractions in their solution.
Brouncker produced rational
solutions which led to arguments.
Frenicle de Bessy was perhaps the only mathematician at that time who
was really interested in number theory but he did not have sufficient
mathematical talents to allow him to make a significant contribution.

Fermat posed further problems, namely that the sum of
two cubes cannot be a cube (a special case of Fermat's Last Theorem which may
indicate that by this time Fermat realised that his proof of the general result
was incorrect), that there are exactly two integer solutions of x2 +
4 = y3 and that the equation x2 + 2 = y3 has
only one integer solution. He posed problems directly to the English. Everyone
failed to see that Fermat had been hoping his specific problems would lead them
to discover, as he had done, deeper theoretical results.

Around this time one of Descartes' students was collecting his correspondence for
publication and he turned to Fermat for help with the Fermat - Descartes correspondence. This led Fermat to
look again at the arguments he had used 20 years before and he looked again at
his objections to Descartes' optics. In
particular he had been unhappy with
Descartes' description of refraction of light and he now settled on a
principle which did in fact yield the sine law of refraction that Snell and
Descartes had proposed. However Fermat had now deduced it from a
fundamental property that he proposed, namely that light always follows the
shortest possible path. Fermat's principle, now one of the most basic
properties of optics, did not find favour with mathematicians at the time.

In 1656 Fermat had started a correspondence with Huygens. This grew out of Huygens interest in probability and the
correspondence was soon manipulated by Fermat onto topics of number theory.
This topic did not interest Huygens but
Fermat tried hard and in New Account of Discoveries in the Science of Numbers
sent to Huygens via Carcavi in 1659, he revealed more of his
methods than he had done to others.

Fermat described his method of infinite descent and
gave an example on how it could be used to prove that every prime of the form 4k + 1 could be written as
the sum of two squares. For suppose some number of the form 4k + 1 could not be
written as the sum of two squares. Then there is a smaller number of the form
4k + 1 which cannot be written as the sum of two squares. Continuing the
argument will lead to a contradiction. What Fermat failed to explain in this
letter is how the smaller number is constructed from the larger. One assumes
that Fermat did know how to make this step but again his failure to disclose
the method made mathematicians lose interest. It was not until Euler took up these problems that the
missing steps were filled in.

Fermat is described in as

Secretive and taciturn, he did not like to talk about
himself and was loath to reveal too much about his thinking. ... His thought,
however original or novel, operated within a range of possibilities limited by
that [1600 - 1650] time and that [France] place.

Carl B Boyer, writing in, says:-

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' 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.