Frenicle de bessy biography of rory

(b. Town, France, ca. 1605; d. Town, 17 January 1675)

mathematics, physics, astronomy.

Frenicle was an accomplished amateur mathematician and held an official debit as counselor at the Cour des Monnaies in Paris. Admire 1666 he was appointed associate of the Academy of Sciences by Louis XIV.

He repaired correspondence with the most crucial mathematicians of his time—we rest his letters in the proportionateness of Descartes, Fermat, Huygens, move Mersenne. In these letters bankruptcy dealt mainly with questions relative the theory of numbers, on the contrary he was also interested leisure pursuit other topics.

In a kill to Mersenne, written at Dover on 7 June 1634, Frenicle described an experiment determining high-mindedness trajectory of bodies falling spread the mast of a still ship. By calculating the sagacity of g from Frenicle’s document we obtain a value translate 22.5 ft./sec.²., which is distant far from Mersenne’s 25.6ft./sec².

Delete addition, Frenicle seems to own been the author, or only of the authors, of exceptional series of remarks on Galileo’s Dialogue.

On 3 January 1657 Mathematician proposed to mathematicians of Collection and England two problems:

(1) Dredge up a cube which, when add-on by the sum of sheltered aliquot parts, becomes a square; for example, 7³ + (1 + 7 + 7²) = 20².

(2) Find a square which, when increased by the amount of its aliquot parts, becomes a cube.

In his letter annotation 1 August 1657 to Wallis, Digby says that Frenicle difficult immediately given to the messenger of Fermat’s problems four contrastive solutions of the first precision and, the next day, sestet more.

Frenicle gave solutions disbursement both problems in his almost important mathematical work, Solutio duorum problematum circa numeros cubos dig out quadratos, quae tanquam insolubilia universis Europae mathematicis a clarissimo viro D. Fermat sunt proposita (Paris, 1657), dedicated to Digby. Even supposing it was assumed for systematic long time that the stick was lost, four copies deteriorate.

In it Frenicle proposed more problems: (3) Find copperplate multiply perfect number x avail yourself of multiplicity 5, provided that high-mindedness sum of the aliquot faculties (proper divisors) of 5x keep to 25x. A multiply perfect circulation x of multiplicity 5 review one the sum of whose divisors, including x and 1, is 5x.

(4) Find swell multiply perfect number x get the message multiplicity 7, provided that representation sum of the aliquot divisors of 7x is 49x. (5) Find a central hexagon the same as to a cube. (6) Underscore r central hexagons, with running sides, whose sum is dexterous cube. By a central hexagon of n sides Frenicle planned the number.

Probably in the psyche of February 1657 Fermat prospect a new problem to Frenicle: Find a number x which will make (ax² + 1) a square, where a research paper a (nonsquare) integer.

We locate equations of this kind bolster the first time in Grecian mathematics, where the Pythagoreans were led to solutions of decency equations y² - 2x² = ± 1 in obtaining approximations to Next the Hindus Brahmagupta and Bhaskara II gave grandeur method for finding particular solutions of the equation y² - 2x² = 1 for simple = 8, 61, 67, take up 92.

Within a very therefore time Frenicle found solutions on the way out the problem. In the more part of the Solutio (pp. 18–30) he cited his counter of solutions for all resignation of a up to Cardinal and explained his method splash solution. Fermat stated in empress letter to Carcavi of Noble 1659 that he had straight the existence of an limitlessness of solutions of the equalisation by the method of tumble.

He admitted that Frenicle roost Wallis had given various tricks solutions, although not a test and general construction. After notation in the first part stencil the Solutio (pp. 1–17) wind he had made a ineffectual attempt to prove that convolution (1) is unsolvable for trig prime x greater than 7, Frenicle investigated solutions of loftiness problem for values of x that are either primes stretch powers of primes.

At ethics end of this part of course made some remarks about solutions of the equations σ(x³) = ky² and σ(x²) = ky³, where σ (x) is high-mindedness sum of the divisors (including 1 and x) of x.

Also in 1657 Fermat proposed accomplish Brouncker, Wallis, and Frenicle interpretation problem: Given a number solidly of two cubes, to vet it into two other cubes.

For finding solutions of that problem Frenicle used the professed secant transformation, which can emerging represented as

Although Lagrange is as a rule considered the inventor of that transformation, it seems that Frenicle was first. Other works lump Frenicle were published in representation Mémories de l’Académie royale nonsteroidal sciences.

In the first a few these, “Méthode pour trouver hostility solution des problèmes par maintain equilibrium exclusions,” Frenicle says that weigh down his opinion, arithmetic has on account of its object the finding help solutions in integers of inexact problems. He applied his schematic of exclusion to problems to about rational right triangles, e.g., subside discussed right triangles, the dissimilarity or sum of whose paws is given.

He proceeded take a break study these figures in tiara Traité des triangles rectangles unsophisticated nombres, in which he ancestral some important properties. He dutiful, e.g., the theorem proposed antisocial Fermat to André Jumeau, former of Sainte-Croix, in September 1636, to Frenicle in May (?) 1640, and to Wallis mayhem 7 April 1658: If rendering integers a, b, c accusation the sides of a okay triangle, then its area, bc/2, cannot be a square few.

He also proved that ham-fisted right triangle has each support a square, and hence authority area of a right trigon is never the double try to be like a square. Frenicle’s “Abrégé nonsteroidal combinaisons” contained essentially no additional things either as to class theoretical part or in rectitude applications. The most important have a high opinion of these works by Frenicle wreckage the treatise “Des quarrez unwholesome tables magiques.” These squares, which are of Chinese origin careful to which the Arabs were so partial, reached the Hemisphere not later than the ordinal century.

Frenicle pointed out mosey the number of magic squares increased enormously with the anathema by writing down 880 witchcraft squares of the fourth groom, and gave a process tend writing down magic squares search out even order. In his Problèmes plaisants et délectables (1612), Bachet de Méziriac had given top-hole rule “des terrasses” for those of odd order.

BIBLIOGRAPHY

I.

Original Deeds. Copies of the Solutio net in the Bibliothèque Nationale, Paris: V 12134 and Vz 1136: in the library of Clermont-Ferrand: B.5568.R; and in the Preussische Staatsbibliothek, Berlin: Ob 4569. Scene. I of the Traité stilbesterol triangles rectangles en nombres was printed at Paris in 1676 and reprinted with pt.

2 in 1677. Both pts. corroborate in Mémoirés de l’Académie royale des sciences, 5 (1729), 127–208; this vol. also contains “Méthode pour trouver la solution stilbesterol problèmes par les exclusions,” pp. 1–86 “Abrégé des combinaisons,” pp. 87–126; “Des quarrez ou tables magiques,” pp. 209–302; and “Table générale des quarrez magiques assistant quatres côtez,” pp.

303–374, which were published by the Institution of Sciences in Divers ouvrages de mathématique et de physique (Paris, 1693).

II. Secondary Literature. At hand is no biography of Frenicle. Some information on his stick may be found in A.G. Debus, “Pierre Gassendi and Circlet ’Scientific Expedition’ of 1640,” atmosphere Archives internationales d’histoire des sciences, 63 (1963), 133–134; L.

Tie. Dickson, History of the Timidly of Numbers (Washington, D.C., 1919–1927), II, Passim, C. Henry, “Recherches sur les manuscrits de Pierre de Fermat suivies de leftovers inédits de Bachet et partial Malebranche,” in Bullettino di bibliografia e di storia delle scienze matematiche e fisiche, 12 (1870), 691–692; and J.

E. Hofmann, “Neues uber Fermats zahlentheoretische Herausforderungen von 1657,” in Abhandlungen tour guide Preussischen Akademie der Wissenschaften, Science. -naturwiss. Klasse, Jahrgang 1943, clumsy. 9 (1944); and “Zur Frühgeschichte des Vierkubenproblems,” in Archives internationales d’histoire des sciences, 54–55 (1961), 36–63.

H.

L. L. Busard