CHINESE MATHEMATICS 599 



is this general form to which we give the name of Diophantine analysis, 

 although Diophantus probably lived after Sun-tsii. One of his prob- 

 lems is as follows : " Find a number which when divided by 3 leaves a 

 remainder of 2 ; when divided by 5 leaves a remainder of 3 ; and when 

 divided by 7 leaves a remainder of 2." At a considerably later date 

 such problems were common in Europe, and were evidently imported 

 from the East. 



Tsu Ch'ung-chih (428-499)^ certainly deserves mention if any 

 standing is to be accorded to Metius in the history of mathematics, 

 since he discovered the latter's value of tt some twelve centuries before 

 it saw light in Europe. About two hundred years before him Liu Hui^ 

 (in 263 A.D.) had given the value ^^Vso (=3.14), and Wang Fan had 

 suggested i^%5 (=3.1555 . . . ). But Tsu Ch'ung-chih, working 

 from inscribed and circumscribed polygons exactly as Archimedes had 

 done, showed that the ratio lay between 3.1415926 and 3.1415927. As 

 limits he fixed upon -y^, the Archimedes superior limit, and ^'^^ig, 

 the value found by Metius. How Tsu came upon these limits we do not 

 know, since his work (the Chui-shu) is lost, but it is possible, as Wei® 

 asserts, that he knew something of infinite series. 



Wang Hs'iao-t'ung, who lived in the first part of the seventh cen- 

 tury, wrote the Ch'i-ku Suan-ching, in which appeared an approximate 

 method of solving a numerical cubic equation. At a later period this 

 would not be significant, but when we bear in mind that this is two 

 centuries before Al Khowarazmi (c. 825) wrote the first book bearing 

 the title " Algebra," and some three hundred years before Alkhazin 

 (c. 950) and Al Mohani were working on this simple cubic, it is in- 

 teresting. 



The golden era of native Chinese algebra was the thirteenth cen- 

 tury, made notable by reason of the works of three men living in widely 

 difierent parts of the empire. Of these, one was Ch'in Chiu-shao,^° 

 who wrote the Su-shu Chiu-chang in 1247. This must always stand out 

 in the history of mathematics as a noteworthy contribution, for here 

 we find the detailed solution of a numerical higher equation by the 

 method rediscovered by Horner in 1819, the only essential difference 

 being in the numerals employed. As already stated, Ch'in merely elab- 

 orated the process for finding the square and cube roots as laid down in 

 the Chiu-chang Suan-shu some fourteen centuries earlier, and this 

 raises the question. How did Leonardo Fibonacci of Pisa solve the 

 numerical equation of which he gives the root to such a high degree 

 of approximation? He wrote his work in 1202. Did he have some 



* Not the sixth century, as Cantor states. 



* Lew-hwTiy. 



* The Chinese historian of mathematics, 

 ^^ Tsin Kiu-tschau, as Cantor has it. 



