# Pierre de Fermat (13934-1)

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**Pierre de Fermat**

Born: 17 Aug 1601 in Beaumont-de-Lomagne, France

Died: 12 Jan 1665 in Castres, France

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

x^{n}
+ y^{n}
= z^{n}

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:-

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