60 HISTORY OF THE THEORY OF NUMBERS. [CHAP. II 



(say 2, 3, 4) obtained by dividing it by 3, 5, 7: 



2-70 + 3-21 + 4-15 = 263, 263 - 2-105 = 53 = ans. 



Similarly for the number not exceeding 315, given the remainders upon 

 division by 5, 7, 9: the remainders are to be multiplied by 126, 225, 280, 

 and from the sum of the products is to be subtracted a multiple of 315. 



Ch'in Chiu-shao 84 gave a method applicable to the problem to find a 

 number x having the remainders TI, , r n when divided by Wi, , m n , 

 which are relatively prime in pah's. Let M be any one of the quotients 

 Mk = Wi 'm A /m k , and seek p so that Mp = 1 (mod m = m k ). We may re- 

 place M by its residue R modulo m. On dividing m by R, let the quotient be 

 Qi and the positive remainder be TI ^ R. Divide R by TI to get the quotient 

 Qz and positive remainder r 2 = r\', divide r\ by r 2 to get the quotient Qs 

 and remainder =i r 2 ; proceed until we reach an r* = 1. Let AI = Q lf 

 A z = AiQ 2 + 1, A 3 = A 2 Q 3 + AI, At, = A 5 Qi + A z , . Then p = A { , 

 and x = riMipi + r 2 M 2 p z + + r n M n p n . 



A German MS. 85 of the fifteenth century proved a general rule corre- 

 sponding to the Chinese t'ai-yen rule. 



Regiomontanus 86 (1436-1476) proposed in a letter the problem to find 

 a number with the remainders 3, 11, 15 when divided by 10, 13, 17. It is 

 possible 87 that he got acquainted in Italy with the work of L. Pisano. 



Elia Misrachi 88 (1455-1526) reproduced L. Pisano 82 (pp. 281-2) and 

 gave answers to similar problems. 



Michael Stifel 89 gave the correct result that if x has the remainders 

 r and s when divided by a and a + 1, respectively, then x has a remainder 

 (a -f l)r + a 2 s when divided by a (a + 1). 



Pin Kue 90 treated in 1593 the problem given by Sun-Tsu. 71 



The problem to find a multiple of 7 having the remainder 1 when 

 divided by 2, 3, 4, 5 or 6 was treated also by Casper Ens 91 and Daniel 

 Schwenter. 92 



Frans van Schooten 93 treated the problem to find a multiple of 7 having 

 the remainder 1 when divided by 2, 3 or 5. He used 30& + 1, where 

 k = 3 is chosen so that the number is divisible by 7. He gave what is 

 really the t'ai-yen rule, but attributed it to Nicolaus Huberti; it leads 

 here to the multipliers 3-5-7 = 105, 2-5-7 = 70, 3(2-3-7) = 126, each 

 with the remainder 1 when divided by 2, 3, 5, respectively. 



84 Nine Sections of Math, (about 1247). Cf. Mikami, 71 pp. 65-9. 



86 M. Curtze, Abh. Geschichtc der Math., 7, 1895, 65-7. 



88 C. T. de Murr, Memorabilia Bibl. publ. Norimbergensium et Universitatis Altdorfinae, 

 Pars I, 1786, p. 99. 



87 Cantor, Geschichte der Math., ed. 1, II, 263. 



88 G. Wertheim, Die Arithmetik des E. Misrachi, 1893, ed. 2, 1896, 60-61. 



89 Arithmetica integra, 1544, Book I, fol. 38v. Die Coss Christoffs Rudolffs, Die Schonen 



Exempeln der Coss Durch Michael Stifel Gebessert, Konigsperg, 1553, 1571. 



90 Swan fa tong tsong, Ch. 5, p. 29, MS. in Bibl. Nat. Paris; abstract by E. Biot, Jour. 



Asiatique, (3), 7, 1839, 193-218. 



91 Thaumaturgus Math., Munich, 1636, 70-71. 



92 Deliciae Physico-Math. oder Math.-u. Phil. Erquickstunden, Niirnberg, 1, 1636, 41. 

 98 Exercitationum math, libri quinque, Lugd. Batav., 1657, 407-410. 



