182 YOSHIO MiKABIl 



son, as given in the commentaries on the "Nine Sections," was effected 

 by a sort of integration. 



The finding of the greatest common measure in the "Nine Sections," 

 Sun-Tse's problem of finding the number whose rests are given when 

 divided by 3, 5 and 7, together with its solution, and Chang Ch'iu-chien's 

 5Mir|5^ "problem of a hundred fowls," — all these appear to be based 

 on the Chinese method of calendrical calculations. The use of the 

 indeternjinate equation 



ax + by = 1, 

 whicli became very prominent in the 13th century, is said to have 

 orginated with the calendrical njethod of I-Hsing — ff (683 727). 

 But before him, its use was certainly implied in the solution of Sun-Tse's 

 problem. 



The method of differences is usually attributed to Kuo Shou-ching 

 |[S^^ (1231-1316). who used three differences. But the case of two 

 differences originated with Pien Kang ^^ long before Kuo (in 892). 

 As mentioned above, however, Tsu Chung-chih and Liu Hui may 

 perhaps have used the method still earlier, 



6. After I-Hsing's time, the study of mathematics long remained 

 at low tide. In the 11th century, at the time of the ascendency of the 

 Stmg philosophy ^'f^, there was a false dawn seeming to herald further 

 progress in the science, but it soon disappeared. It was not until the 

 13th and 14th centuries that Chinese mathematics underwent a brilliant 

 development. Some sixty years during this period may be said to 

 constitute the golden age of Chinese mathematics. The algebraical 

 method known as U-ten-yuen-i-shu 'it3^jt~^^^ or temjenjutsu '^jtM^ i^ 

 Japanese, appeared diu'ing this period, and higher numerical equations 

 were ajjproximately solved by means of calctilating sticks, and according 

 to a principle that may be compared with Horner's method. But it was 

 no other than a direct extension of the root extracting operations of old 

 times to the case of higher equations, the operations being carried out 

 with calculating sticks. Arrangements of algebraical expressions were 

 also made by the same method. 



Chu Shih-chieh ;^ifi:'^ discovered the szu-yuen-sliu ^jt'^^^, or 

 "the method of four elements," at the beginning of the 14th century. 

 In the U-teii-yuen-i-sliu, one element or one unknown only was contained 

 in an equation. But in Chn's method, there were admitted elements 

 or uidcnowus up to four in number, the four elements being arranged at 

 tlie foin- sides around the absolute term wliich was placed in the middle. 



