Trait? des triangles rectangles en nombres, dans lequel plusieurs belles propriet?s de ces triangles sont d?montr?es par de nouveaux principes

FRENICLE DE BESSY, Bernard

Edité par Estienne Michallet, Paris, 1676

A propos de cet article

THE FIRST CASE OF FERMAT?S LAST THEOREM TO BE PROVED HENRY CAVENDISH?S COPY. First edition, very rare, Henry Cavendish?s copy, of this early treatise on number theory, containing results Frenicle had probably obtained by correspondence with Fermat, including the proof of the first case of Fermat?s last theorem to be published. ?During the first half of the seventeenth century, mathematical ideas circulated in France, and to some extent throughout Western Europe, via a correspondence network, having its centre in Paris, and with Marin Mersenne (1588-1648) as its driving force. Pierre de Fermat (1601?-1665), lawyer and mathematician living near Toulouse in the south of France, joined this circle in 1636. Bernard Frenicle de Bessy (1605?-1675), astronomer, physician, naturalist, and mathematician from Paris, had first communicated with Mersenne in 1634. It seems to have been mid-1640, however, before they [i.e., Fermat and Frenicle] were corresponding directly with each other, although both parties were using Mersenne as an intermediary earlier that year. Fermat and Frenicle were each interested in the theory of numbers, and in questions concerning aliquot parts (divisors of positive whole numbers). Indeed, as early as 26th December 1638, Fermat had written to Mersenne claiming that he could solve by his method all questions concerning aliquot parts, but in his usual way he had omitted any details. These claims no doubt aroused the interest of Frenicle, and at the beginning of 1640 he wrote to Mersenne with a question for Fermat ? This challenge from Frenicle was the beginning of a correspondence between the two men? (Fletcher). Correspondence with a man of Fermat?s genius was obviously of great benefit to Frenicle, but Fermat was also provoked into new discoveries by Frenicle. ?Fermat, in a letter to Gilles Personne de Roberval, writes, ?For some time M Frenicle has given me the desire to discover the mysteries of numbers, an area in which he his highly versed?? (MacTutor). In 1643 Fermat wrote to Pierre de Carcavi, ?There is certainly nothing more difficult than this in the whole of mathematics and, except for M Frenicle and perhaps for M Descartes, I doubt if anyone understands the secret? (ibid.). Moritz Cantor noted that Frenicle?s expertise should not be judged solely on the basis of his few published works. ?Further research by Freniclemust have been known confidentially, because, according to the work published by the Academy, the great esteem that Fermat, in particular, devoted in Frenicle cannot be explained in any other way? (Goldstein, p. 427).In his Trait? des triangles, Frenicle proved that, ?if the integers a, b, c [with a being the hypotenuse] represent the sides of a right triangle [a, b, c being integers (whole numbers)], then its area, bc/2, cannot be a square number. He also proved that no right triangle has each leg a square, and hence the area of a right triangle is never the double of a square? (DSB). The statement that no right triangle has each leg a square is the case n = 4 of Fermat?s last theorem, that the sum of two nth powers cannot be an nth power if n is greater than 2. Frenicle?s proof is by the ?method of descent?, which was invented by Fermat and which Frenicle very probably learned through correspondence with him. Frenicle?s Trait? des triangles was reprinted, together with a second part, in 1677 in the Recueil de plusieurs Trait?z de Math?matique de l'Acad?mie Royale des Sciences. OCLC lists 11 copies worldwide (Brown, Columbia, and Harvard in the US). RBH lists only the Macclesfield copy since the present copy sold in 1980. Provenance: Henry Cavendish FRS (1731-1810), natural philosopher (his ink stamp on title verso). Cavendish was ?the greatest experimental and theoretical English chemist and physicist of his age. Cavendish was distinguished for great accuracy and precision in research into the composition of atmospheric air, the properties of different gases, the synthesis of water, the la.

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Détails bibliographiques

• Titre: Trait? des triangles rectangles en nombres, dans lequel plusieurs belles propriet?s de ces triangles sont d?montr?es par de nouveaux principes
• Éditeur: Estienne Michallet, Paris
• Date de parution: 1676
• Édition: First edition.
• Catalogues du vendeur:
  • Geometry, Number Theory
  • Mathematics, Analysis, Algebra