350 HISTORY OF THE THEORY OF NUMBERS. [CHAP, xn 



The next step gives 0" = - 7; the third, $"' = - 2. His solution of 

 x z Qly 2 = 1 is quoted by Whitford 4 (pp. 37-8), who remarked that the 

 wording is clearer than in Colebrooke's translation. 



El-Hassar 32 (1432) obtained for Va 2 + r, when a = 2, r = I, the approxi- 

 mations a + p = 9/4, where p = r/(2a), and 



a + p- P 2 /{2(a + p)} = 161/72. 



[Note that (9, 4) and (161, 72) are solutions of z 2 - - 5y- = 1.] 



Nicolas Chuquet 33 obtained, in 1484, successive approximations to Vn 

 for n ^ 14. He began by noting that V6 lies between 2 and 3. Their 

 arithmetical mean is 2|; its square 6| exceeds 6 by \. Take the next 

 smaller term f in the series , |, i, |, . We have 2f , whose square is 

 less than 6. We now have an approximation exceeding the root and one 

 less than it. Adding the numerators and denominators of | and ^, we get 

 the new approximation 2f , whose square < 6. Similarly from 2| and 2f 

 we get 2f. In this way he obtained the approximations 2 + r, where 



1 1 2. 3. 4. _5_ 9 13 22 31 4J) 49 _8 9_ 

 ' ' 1i 3 > 5 ) 7 ) 9 ) 11) 20) 29) 49) 69) 89) 109) 198' 



[For r = 0, ^, f, ^o, -ft, j^, 2 + r gives the successive convergents to the 

 continued fraction for V6. To deduce a third convergent p^/qz from two 

 successive ones Po/qo, Pifqi, the law is pz = PO + zpi, #2 = qo + ^gi. Thus 

 Chuquet's process produced also intermediate fractions, obtained by re- 

 placing z by smaller numbers.] Chuquet 34 gave answers to the following 

 problems, but with no details as to solution. Find a square which increased 

 by 7 (or 4) gives a square; answer, 9 (or 9/4). Find three squares whose 

 sum is 13; answer, 11-J, If, --. Find three cubes whose sum is 20; answer, 



1 f^S. 03. 1 

 1O 8> 8> Lm 



Jordanus Nemorarius 35 noted that x(x + 1) is neither a square nor a 

 cube [if x is an integer 4= 0, 1 ; for x = %, it equals (f ) 2 ]. 



Estienne de la Roche 36 copied the above method of approximation from 

 Chuquet's manuscript. 



Juan de Ortega in the later editions (1534, 1537, 1542) of his Arithmetica 

 gave the approximations 



Vl28 = llM, ^297 = 17,Mro, ^300 = 17f|, 



V375 = 19fH, ^135 = llH, 

 which correspond 37 to the first solution of x 2 Dy 2 = 1, and 



V80 = 8i, -V75 = $m, ^756 = 27MS, ^231 = l&Hft, 

 which correspond to the second solution. 



32 H. Suter, Bibliothcca Math., (3), 2, 1901, 37. Also simultaneously by Alkalgadi, French 



transl. in Atti Accad. Pont. Nuovi Lincei, 12, 1858-9, 402^1. 



33 Le triparty en la science des nombres, Bull. Bibl. Storia Sc. Mat., 13, 1880, 697-9. Dis- 



cussed by S. Giinther, Zeitschrift fur das Realschulwesen, 2, 1877, 430; L. Rodet, Bull. 



Soc. Math, de France, 7, 1879, 162; P. Tannery, Bibliotheca Math., (2), 1, 1887, 17. 

 84 Le triparty . . . , Appendix; Bull. Bibl. Storia Sc. Mat., 14, 1881, 455. 

 86 Elementa Arith. decem libris, demonstr. Jacobi Fabri Stapulensis, Paris, 1514, VI, 26. 



86 Larismetique, 1520. 



87 J. Perott, Bull. Bibl. Storia Sc. Mat. Fis., 15, 1882, 169. Cf. P. Tannery, Bibliotheca Math., 



(2), 1, 1887, 19-20. 



