MATHEMATICS LV CHINA AND JAPAN. 181 



of quadratic equations. lu the 7th century, Wang Hsiao-t'ung 2E# jl, 

 solved various problems by means of cubic equations, whose solutions, 

 it is implied, were effected in the same way as in the cubic root-extraction, 

 as the technical terms employed sliow very sufiSciently. In tlie "Nine 

 Sections," linear simultaneous equations are solved very ingeniously. 

 Liu Hui's §lj^ commentaries, composed in 263 A. D., contain a new 

 method of solving them. It is notable that these operations of root 

 extraction and also of solving linear systems were executed by means 

 of the calculating sticks, which consisted of two sorts, red and black, 

 representing positive and negative numbers. Without noting this lact, 

 it will be hard to understand the real nature of the Chinese system of 

 algebra. Its later development was to follow the same course. 



Though important for tlie historical study of arithmetic and algebra, 

 yet the algebraical methods employed in these calculations were chieriy 

 built up on a geometrical basis, as is apparent from the commentaries 

 on the older works. The arithmetical root-extraction as well as the 

 treatment of quadratic equations were both explained by the use of 

 geometrical figures. 



The cubature of some solids was tried with considerable care. In tlie 

 quadrature of the circle, a considerably high degree of logical exactness 

 was observed. Liu Hui inscribed a 96-gon witiiin a circle and tried a 

 kind of correction upon the values obtained lor successively ^inscribed 

 polygons, doubling each time the number of their sides up to the 96-gou, 

 so as to get the value for the circle. Tsu (Jh'ung-chih's jfllffp^ 

 (429-500) method of quadrature is not thorougnly understood, because 

 his writings are not extant; but we may not luijustly guess that he 

 used a method like the chao-clta method ^^^j^ or 'the method 

 of differences," wiiich has developed in subsequent years. Liu Hui's 

 method of correction was certainly the starting point for the chao-clia 

 method. The commentaries on the "Nine ISections" and the short 



article in the Sui Shu ^^ (History of the Sui Dynasty, 589 to 618 

 A. D.) on Tsu's quadrature seem clear when compared with the works 

 of i^eki Kowa f^^^O and other Japanese mathematicians. The early 

 Japanese method of quadrature, as described in the KioaUuyo Soyinpo 

 ^l^lc^^i? may closely resemble that used by Tsu. The latier's fi-.ictional 

 value of ;r=355/113 was probably obtained by a process similar to 

 that used by fSeki. In this case, Tsu's treatment would have been 

 decidedly algebraical, deviating i'rom the Greek geouietrical method. 

 The cubatiu-e of a sphere by Tsu llcng- chili j]ilill'j'3;^ , another Tsu's 



