**Frenicle de Bessy** was a French amateur mathematician who is best known for his contributions to number theory.

- Very little is known about his life (even his year of birth is a guess) and, given that he was one of the founder members of the Paris Academy of Sciences and has an Éloge written by the Marquis de Condorcet, this must give us one obvious fact about Frenicle de Bessy, namely that he was a very private man.
- Even Pierre de Fermat, with whom Frenicle de Bessy corresponded, commented in August 1638 that he knew nothing of the man.
- Historians have come up with some guesses about Frenicle de Bessy, but little in the way of hard facts about him have emerged other than his letters and writings about mathematics.
- It exercised important advisory and administrative functions, helping the government periodically to fix the value in livres, sous and deniers of the many types of coinage in France, and being responsible for drafting royal edicts on financial affairs.
- It oversaw the management and output of the thirty mints which operated in the kingdom, to which end it despatched its 'conseillers' on special missions.
- This was the environment in which Frenicle de Bessy spent much of his time.
- He subsequently joined the Montmor and Thévenot 'academies', assisting from time to time in astronomical observations conducted by members of the latter group.
- when names were being canvassed for the Académie des Sciences, that of Frenicle de Bessy was among those regarded as most likely to be included.
- If anyone could help the new institution to work harmoniously it was Frenicle de Bessy.
- He corresponded with René Descartes, Pierre de Fermat, Christiaan Huygens and Marin Mersenne.
- Most of the correspondence between these men and Frenicle de Bessy was on number theory but not exclusively so.
- In a letter which he wrote at Dover in England to Mersenne on 7 June 1634, Frenicle describes an experiment to study the trajectory of a body released from the top of the mast of a moving ship.
- Again on a more applied mathematical topic, Frenicle wrote an article which makes comments on Galileo's Dialogue.
- However, the famous mathematical historian Moritz Cantor felt that since Frenicle was so highly regarded by other mathematicians of his day that he must have produced further research which was known to his colleagues at the time but it was never published and no record of it has come down to us.
- It is interesting to look at a comment about Frenicle in a letter of one of his correspondents.
- Digby clearly knew Frenicle well and several letters from Frenicle to Digby around 1658 are extant.
- All of this suggests that Frenicle did not have as good a mathematical background as he might have had, so his talent must have been in possessing amazing computational skills.
- It was this remarkable computational ability that means that today Frenicle de Bessy is best known for his contributions to number theory.
- He solved many of the problems posed by Fermat but he did more than find numerical solutions for he also put forwards new ideas and posed further questions.
- Later their missives became more casual and Fermat actually revealed things to de Bessy concerning his mathematical methods that he refused to divulge to his other correspondents.
- We shall look at some of the problems which were typical of those Frenicle worked on.
- On 3 January 1657 Fermat made a challenge to the mathematicians of Europe and England.
- Frenicle solved other problems posed by Fermat.
- We note that the Solutio is the only publication of Frenicle in his lifetime although other memoirs by him were published after his death.
- Using his great skill in combinatorial mathematics and in computation, Frenicle de Bessy worked on magic squares.
- His two memoirs Des quarrez magiques Ⓣ(Magic squares) and Table générale des quarrez magiques de quatre de côté Ⓣ(General table of 4 x 4 magic squares) were published in 1693, nearly 20 years after his death.
- In this work he listed 880 magic squares of order 4.
- In fact, this is the complete list of magic squares of order 4 but Frenicle's papers do not prove this.
- It appears that a proof that there were exactly 880 magic squares of order 4 did not appear until 1931 when Friedrich Fitting (1862-1945) published the paper Rein mathematische Behandlung des Problems der magischen Quadrate von 16 und von 64 Feldern Ⓣ(Pure mathematical treatment of the problem of magic squares of 16 and 64 squares).
- Frenicle also gave methods to find magic squares of any even order.
- These memoirs by Frenicle were two of four published in Divers ouvrages de mathématique et de physique Ⓣ(Various mathematical and physical works) (1693).
- M de la Hire examined all the manuscripts that he had chosen ...
- The other two manuscripts by Frenicle that were published in this volume were Methode pour trouver la solution des problèmes par les exclusions Ⓣ(Method for the solution of problems by elimination) and Abregé des combinaisons Ⓣ(Abstract combinations).
- These were chosen by de la Hire to be the first two in the published collection.
- He joined a treatise on 'Combinations', and then he decided that it was necessary to leave for another time several other works by M Frenicle, which all together would have made a very large volume, such as papers on prime numbers, another on polygonal numbers, one of tables of magic squares, and others: but to make it a more perfect volume, he added papers on magic squares; and he believed that the public would be glad to see that what had been published up to then by the ablest algebraists, was far removed from what M Frenicle had discovered on this matter.
- We note that Frenicle's Methode pour trouver la solution des problèmes par les exclusions Ⓣ(Method for the solution of problems by elimination) presents ten rules which he suggests are useful in solving mathematical problems.
- In particular he looks at finding right angled triangles when the difference or the sum of two of the sides are given.
- In many ways these rules emphasise the point that we made earlier about Frenicle being primarily a remarkable calculator, for these rules give essentially an experimental approach to finding integer solutions to specific number theory problems.
- As we mentioned at the beginning of this article, Frenicle was elected as a founder member of the Académie Royale des Sciences in 1666.

Born about 1604, Paris, France. Died 1674, Paris, France.

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Algebra, Ancient Indian, Astronomy, Number Theory, Special Numbers And Numerals

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Parts: 2

Sections: 3

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