**Pierre de Fermat** was a French lawyer and government official most remembered for his work in number theory; in particular for Fermat's Last Theorem. He is also important in the foundations of the calculus.

- 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.
- 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.
- 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.
- Of course Fermat was preoccupied with mathematics.
- 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.
- 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.
- 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.
- 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.
- 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 Ⓣ(Dioptrics) 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.
- 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 méthode Ⓣ(Discourse on Method) alone could give.
- Descartes attacked Fermat's method of maxima, minima and tangents.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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+14k + 14k+1 could be written as the sum of two squares.
- 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.
- 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.

Born 17 August 1601, Beaumont-de-Lomagne, France. Died 12 January 1665, Castres, France.

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Algebra, Analysis, Ancient Indian, Number Theory, Physics, Puzzles And Problems, Special Numbers And Numerals

Parts: 1 2

Sections: 3

Solutions: 4 5

Theorems: 6 7

**O’Connor, John J; Robertson, Edmund F**: MacTutor History of Mathematics Archive