MATHEMATICS IN CHINA AND JAPAN. t93 



It was by the aid of this relation, tliat the problem of the six 

 tangents to four circles touching another was solved, the solution being 

 nothing l)ut Casey's theorem, obtained tens of years ahead of Casey. 

 The relation of the six tangents was in turn used to solve many other 

 problems. Thus there arose various prol)lems in which 'a number of 

 circles were variously inscribed within a larger circle. 



There also appeared a certain method that may remind us of 

 inversion. In this connection, Hasegawa Kwan's ^^jl]^ (1782-1838) 

 Kydkngyo-jnUu ^^^, or "^method of limiting figures," is to be noticed. 

 It was not an accurate method indeed, because it sometimes erred in its 

 conclusions, but it proved to be a step toward further progi-ess. The 

 work of Hodqji Zen -^^^^^ (1820-1868) in establishing a new method 

 of inversion-let us be allowed to use such a term-was based on Hasegawa's 

 method, as he himself acknowledged. Hodoji had no idea of inversing a 

 figure into another, it is true, nor did he think of the center of inversion. 

 But his transformations gave the same results as inversion. Moreover 

 he had the idea of a tangent common to two circles, even when one 

 of them was entirely within the other. By means of this method he 

 was able to solve many a jiroblem. His method, however, was neither 

 printed nor otherwise made public. Before him, Ushijima Seiyo ^% 

 ^M (1756-1840) had certainly discovered a similar method, though his 

 researches are little known. Omura Isshu 'j\1^—^ also gave the same. 

 In considering the development of this sort of treatment, we must take 

 into account, besides the jtrogress of the considerations of variously 

 inscribed circles, also the fact that the geometrical interpretations of 

 the roots of algebraical equations had exercised no small influence. 



In this manner the branch of geometry in Japanese mathematics 

 had been gradually advancing in organization, many noticeable results 

 being estaldished Irom time to time. But now all advancement on a 

 sudden came to a stop, for tlie study ga^^e way to the im])ortation of 

 Western mal hematics on a large scale. 



17. The theory of numbers and the solution of indeterminate 

 ])roblems were also themes fondly attacked by Jai>anese mathematicians. 

 The use of tht; equation 



ax + by = 1 

 was certainly due to China, but it was used in Japan to solve a large 

 nund)er of problems unknown in China. For integral solutions of a 

 right triangle, a number of results were obtained. The oblique triangle 

 was also solved in several ways. There were many other sorts of figures, 



