## Traité des triangles rectangles en nombres, dans lequel plusieurs belles proprietés de ces triangles sont démontrées par de nouveaux principes.

Paris: Estienne Michallet, 1676.

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 *Traité 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 *n*th powers cannot be an *n*th 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 *Traité des **triangles* was reprinted, together with a second part, in 1677 in the *Recueil de plusieurs Traitéz de Mathématique de l'Académie 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 law governing electrical attraction and repulsion, a mechanical theory of heat, and calculations of the density (and hence the weight) of Earth. His experiment to weigh Earth has come to be known as the Cavendish experiment” (Britannica). Cavendish’s private library in Bedford Square, London, “was open to the qualified public, but its contents were not selected with the public in mind. The largest category in the catalog was natural philosophy, with nearly 2000 titles … Mathematics, the second largest category, included in addition to books on pure mathematics, books on natural philosophy in which mathematics was used, such as Newton’s *Principia *and *Opticks *and Robert Smith’s *System of Opticks*. Astronomy was a category of its own and well represented, including classic works of science by Copernicus, Brahe, Kepler, and others” (Jungnickel & McCormmach, pp. 275-276). Upon his death, Cavendish’s library passed to Lord George Cavendish, who gifted it to his nephew, the Sixth Duke of Devonshire. The vast majority of the books remain in the library at Chatsworth, although a few have reached the market (for example, Horblit’s copy of Leybourn’s *Cursus mathematicus* (1690), as well as the present work).

“Perhaps none of Fermat’s contemporaries deserves to be called a number-theorist; but several were number-lovers. One of these was Mersenne, who was interested in ‘perfect’ and ‘amicable’ numbers; so were P. Bruslart de Saint-Martin and A. Jumeau de Sainte-Croix, both older than Fermat, and B. Frenicle de Bessy, who was some years younger and was to become a member of the first Academy of Sciences when it was founded by Colbert in 1666. Frenicle was once praised by Fermat, not without a touch of irony perhaps, as ‘this genius who, without the succor of Algebra, penetrates so deeply into the knowledge of numbers.’

“Early in his career, in his Bordeaux period, Fermat had been attracted by ‘magic squares’; these are square matrices of positive integers satisfying certain linear conditions. Stifel, Cardano, Bachet had mentioned this topic. In 1640, Fermat and Frenicle discovered through Mersenne their common interest in it; this was the beginning of a correspondence between them, which, after a somewhat inauspicious beginning, soon turned to more substantial topics and, in spite of some occasional friction was continued, rather desultorily at times, throughout the next two decades” (Weil, pp. 51-52).

Fermat communicated some of his most important number-theoretic discoveries to Frenicle. In a letter to Frenicle dated October 18, 1640, Fermat stated (without proof) his so-called ‘little theorem’: if *a* is any integer and *p* is a prime number, then *p* divides *a ^{p}* –

*a*(see Ore, pp. 272-273; Euler was the first to publish a proof). In the same year, Fermat expressed to Frenicle his famous statement about ‘Fermat primes’ (

*ibid*., pp. 73-74). If the integer

*n*is not a power of 2, it is easy to show that the number 2

*+ 1 cannot be a prime number. Fermat stated to Frenicle that the converse of this is true, i.e., that 2*

^{n}*+ 1 is prime if*

^{n}*n*is a power of 2. In this, Fermat was mistaken – Euler proved that 2

^{32}+ 1 is not prime. In the other direction, again in 1640, Frenicle wrote to Fermat with a question about ‘perfect numbers’: these are numbers which are equal to the sum of their divisors (e.g., 28 = 1 + 2 + 4 + 7 + 14). “Frenicle asked Fermat, through Mersenne, for a perfect number between 10

^{20}and 10

^{22}… The point in Frenicle’s question was really to find out whether 2

^{37}

**–**1 is a prime. It is not, and there is no perfect number between 10

^{20}and 10

^{22}(or at any rate no ‘euclidean’ one), as Fermat answers soon enough; fortunately for his self-esteem, he had detected, just in time, a numerical error in his calculations, which had nearly made him fall into the trap carefully laid for him by Frenicle” (Weil, pp. 54-55).

“In his *Traité des **triangles**rectangles en nombres, *posthumously published in 1676, Frenicle did include a proof that the area of a pythagorean triangle [i.e., a right-angled triangle the lengths of whose sides are integers] can neither be a square nor twice a square … one may safely assume that this was based on a communication from Fermat …

“Writing to Huygens about his proof by descent for the theorem on the area of pythagorean triangles, Fermat merely described the method by saying that ‘if the area of such a triangle were a square, then there would also be a smaller one with the same property, and so on, which is impossible’; he adds that ‘to explain why would make his discourse too long, as the whole mystery of his method lies there’ …

“Take a pythagorean triangle whose sides may be assumed mutually prime [i.e., they have no common factor]; then they can be written as (2*pq*, *p*^{2 }– *q*^{2}, *p*^{2 }+ *q*^{2}) where *p, q* are mutually prime, *p* > *q*, and *p *–* q* is odd. Its area is *pq*(*p + q*)(*p *–* q*), where each factor is prime to the other three; if this is a square, all the factors must be squares. Write *p = x*^{2}, *q = y*^{2}, *p + q = u*^{2}, *p *–* q = v*^{2}, where *u*, *v* must be odd and mutually prime. Then *x*, *y* and *z = uv* are a solution of *x*^{4} – *y*^{4} = *z*^{2} … We have *u*^{2} = *v*^{2} + 2*y*^{2}; writing this as 2*y*^{2} = (*u* + *v*)(*u *–* v*), and observing that the greatest common divisor of *u + v* and *u *–* v* is 2, we see that one of them must be of the form 2*r*^{2} and the other of the form 4*s*^{2}, so that we can write *u* = *r*^{2} + 2*s*^{2}, *v = r*^{2} – 2*s*^{2}, *y* = 2*rs*, and consequently *x*^{2} = ½ (*u*^{2} + *v*^{2}) = *r*^{4 }+ 4*s*^{4}. Thus *r*^{2}, 2*s*^{2} and *x* are the sides of a pythagorean triangle whose area is (*rs*)^{2} whose hypotenuse is smaller than the hypotenuse *x*^{4} + *y*^{4} of the original triangle. This completes the proof ‘by descent’.

“As to triangles whose area is twice a square, he proceeds as follows. As above, one can write *p + q = u*^{2}, *p – q = v*^{2}** , **and either

*p = x*

^{2},

*q*= 2

*y*

^{2}, or

*p*= 2

*x*

^{2},

*q = y*

^{2}. As

*u*,

*v*are odd, and 2

*p*=

*u*

^{2}+

*v*

^{2},

*p*must be odd, and so we can write

*p = x*

^{2},

*q*= 2

*y*

^{2}. Then 4

*y*

^{2}= (

*u + v*)(

*u*–

*v*); as the greatest common divisor of

*u + v*and

*u – v*is 2, we can write

*u + v*= 2

*r*

^{2},

*u*–

*v*= 2s

^{2},

*u = r*

^{2}+

*s*

^{2},

*v = r*

^{2}–

*s*

^{2}, and finally

*x*

^{2 }= ½ (

*u*

^{2 }+

*v*

^{2}) =

*r*

^{4}+

*s*

^{4}. Then the triangle (

*r*

^{2},

*s*

^{2},

*x*) has the area 2(

*rs*/2)

^{2}and the proof is complete.

“From this, as Frenicle observes, and as Fermat must have known, a number of consequences follow. For instance, the fact that the equations *x*^{4} + *y*^{4} = *z*^{2} and *x*^{4} – *y*^{4} = *z*^{2} have no non-trivial solutions is included in the above proofs. Also, in a pythagorean triangle (*a*, *b*, *c*), *a* and *b* cannot both be squares, since then the area *ab*/2 would be twice a square; *a* and *c* cannot both be squares, since *a = r*^{2}, *c = s*^{2} would give *s*^{4} – *r*^{4} = *b*^{2}. From this one can deduce Fermat’s statement (*Oeuvres* I, 341) that no triangular number other than 1 is a fourth power. Triangular numbers are of the form ½ *n*(*n*+1); if this is a fourth power, one of the integers *n*, *n* + 1 must be of the form *x*^{4} and the other of the form 2*y*^{4}, so that we have *x*^{4} – 2*y*^{4} = 1 or –1. If *x *> 1, consider the pythagorean triangle *a = x*^{2}, *b* = ½ (*x*^{4} – 1), *c* = ½ (*x*^{4} + 1); in one case *a *and *b* would be squares, and in the other case *a* and *c* would be such, contradicting what has been proved before. Similarly, by considering the triangle *a = x*^{2}*z*^{2}, *b* = ½ (*x*^{4} – *z*^{4}), *c *= ½ (*x*^{4} + *z*^{4}), one sees that the equations *x*^{4} +* z*^{4} = 2*t*^{2} and x^{4} –* z*^{4} = 2*t*^{2} have no non-trivial solutions” (*ibid*., pp. 76-79).

“Born and raised in Paris, Frenicle de Bessy must have graduated in law before proceeding to hold the office of ‘Conseiller à la cour des monnaies’. This tribunal had been a sovereign court since 1552, which is to say that its writ ran throughout the kingdom, and in certain areas of competence it ranked as a final court of appeal. As its title indicates, it paid particular attention to subjects pertaining to coinage and finance. 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 … It tried both civil and criminal cases concerning forgery, counterfeit, or any dispute over the coinage of the realm ... This was the environment in which Frenicle de Bessy spent much of his time … He may have been a ‘conseiller’ by the late 1630s when he was attending meetings of Mersenne’s group. 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. Not least among his advantages was his cooperative, genial personality … If anyone could help the new institution to work harmoniously it was Frenicle de Bessy” (MacTutor).

“In 1666 [Frenicle] was appointed member of the Academy of Sciences by Louis XIV. He maintained correspondence with the most important mathematicians of his time – we find his letters in the correspondence of Descartes, Fermat, Huygens, and Mersenne. In these letters he dealt mainly with questions concerning the theory of numbers, but he was also interested in other topics. In a letter to Mersenne, written at Dover on 7 June 1634, Frenicle described an experiment determining the trajectory of bodies falling from the mast of a moving ship … In addition, Frenicle seems to have been the author, or one of the authors, of a series of remarks on Galileo’s *Dialogue*” (DSB).

Frenicle, while not a great theoretician like Fermat, was a brilliant calculator. On 3 January 1657 Fermat proposed to the mathematicians of Europe and England the following problem: “to find a cube which, when increased by the sum of its aliquot parts, becomes a square (for example, 7^{3} + (1 + 7 + 7^{2}) = 20^{2}). In his letter of 1 August 1657 to Wallis, Digby says that Frenicle had immediately given four different solutions of the problem and, the next day, six more.

Fletcher, ‘A Reconstruction of the Frenicle-Fermat Correspondence of 1640,’ *Historia Mathematica* 18 (1991), pp. 344-351. Goldstein, ‘L’experience des nombres de Bernard Frenicle de Bessy,’ *Revue de Synthèse* 122 (2001), pp. 426-454. Jungnickel & McCormmach, *Cavendish. The Experimental Life*, 2^{nd} edition, 2016. Ore, *Number Theory and Its History*, 1948. Weil, *Number Theory. An Approach through History from Hammurapi to Legendre*, 1984.

12mo (149 x 91 mm), pp. [iv], 116. Contemporary blind-ruled calf, expertly re-backed, spine gilt in compartments with red lettering-piece. A fine copy.

Item #6170

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Price:
$9,500.00
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