CHAP. XII] PELL EQUATION, az 2 +&z+c= D. 351 



J. Buteo 38 gave several approximations for V66 all of which give solu- 

 tions of x 1 - - QGy 2 = 1, the last one being x/y, x = 8449, y = 1040. He also 

 made use of Chuquet's method. 



P. A. Cataldi 39 gave approximations to V44 by the two formulas used 

 by El-Hassar 32 and used implicitly approximations by continued fractions. 



Nicolas Rhabdas 40 used the first approximation by El-Hassar. It was 

 used later by Luca Paciuolo, Cardan and Tartaglia (references, Vol. I, Ch. I). 



Fermat 41 stated February, 1657, that if D is any number not a square 

 there exists an infinitude of integral solutions of x 2 Dy 2 = 1; for ex- 

 ample, 2 2 -- 3 -I 2 = 1, 7 2 - 3-4 2 = 1. He asked for the least solution of 

 Qly 2 + 1 = D and of 109?/ 2 + 1 = D, and a general rule for finding the 

 solutions of Dy~ + 1 = D . 



Although Fermat, in the introductory remarks to his " Second defi," 

 had expressly called for solutions in integers, this introduction was omitted 42 

 in the copy made for Lord Brouncker by the secretary of K. Digby. This 

 explains why W. Brouncker and John Wallis 43 first gave merely the rational 

 solution 



4ps s 



2 



x = - - 11 == 

 ' 



s 2 4p 2 n' s 2 



of nx~ + 1 = ?/ 2 , the case p = 1, s = 2r, giving Brouncker's solution 

 x = 2r/(r 2 ri). The latter had been given by Bhascara 30 (second method), 

 and was obtained by Rene Francois de Sluse 44 (1622-1685) by setting 

 nx 2 + 1 = (1 - rx}\ 



Fermat 45 was not satisfied with these evident solutions in fractions. 



W. Brouncker 46 gave an infinitude of integral solutions x for n = 2, 3, 

 5, 6 and their products by squares; thus, for n = 2, 



x = 2x5ix5fx5ff X , 



each numerator being equal to the corresponding denominator diminished 

 by the preceding denominator, while each denominator equals the numerator 

 of the term immediately preceding when reduced to an improper fraction. 

 [The formula gives \x 1, 6, 35, 204, 1189, , with the recursion formula 



Wallis 47 noted that if x = f is one solution, so that nf 2 + 1 = I 2 , then 

 x = 2fl is a second: n(2fl} 2 + 1 = (2l 2 I) 2 , so that one can get an 



38 loan. Buteonis Logistica, quae et arith. . . . , Lyons, 1559, 76. 



39 Trattato del Modo Brevissimo di trouare la Radice quadra delli numeri, Bologna, 1613, 12. 



40 P. Tannery, Notice sur les deux arithmetiques de N. Rhabdas, Paris, 1886, 40, 68. 



41 Oeuvres, II, 333-5, letter to Frenicle and " Second defi aux mathematiciens " [Wallis and 



Brouncker]; French transl. of latter, III, 312-3. 



42 G. Wertheim, Abhandl. Geschichte Math., 9, 1899, 563. 



43 Commercium epistolicum de Wallis, Oxford, 1658, 767; bound with Wallis' Algebra, 



Oxford, 1685; Wallis' Opera, Oxford, 2, 1693. French transl. in Oeuvres de Fermat, 

 III, 417-8; letter IX, Wallis to Digby, Oct. 7, 1657. 



44 MS. 10247, f. 286 verso, du fonds latin, Bibliotheque Nat. de Paris. 



45 Oeuvres, II, 342, 377; letters to Digby, June 6, 1657, April 7, 1658. 



46 Commercium, 775, letter XIV, Nov. 1, 1657; Oeuvres de Fermat, III, 423. 



47 Letters XVI, XVIII to Digby, Dec. 1, and Dec. 26, 1657; Oeuvres de Fermat, III, 434-5; 



480-9. 



