MATHEMATICS IN CHINA AND JAPAN, 189 



"metliod of differences" — shosaho I^Hf;^- In the method of ch-cle- 

 measureujent used b)^ tlie Takunm School ^f^:^^, expansions in infinite 

 series were obtained from similar data by the direct application of the 

 method of differences. The writers of the Taisei Sanhyo 'j^}^^^ 

 and other manuscript works employed similar methods and arrived at 

 formulae more convenient for numerical calculations. 



In these considerations some were evidently anticipating expansions 

 in infinite series, still others obviously based their deductions on such 

 expansion. As an example of the mathematical method by which an 

 infinite expansion was openly aimed at, we may refer to the analytical 

 treatment described in the Yenri Tetsvjutsu MMMMh^ '-^ method usually 

 called by the name yGuri or -'circle-principle." It is sometimes believed 

 to be Seki's discovery, but a work of 1726 is the first treatise in which 

 it is described as now known. By this method succesive numerical 

 values are not found, but successive calculations are all made analytically, 

 using binomial expansion, and applying the method of induction, though 

 imperfectly. 



Binomial expansion was executed in exactly the same manner as 

 in the operation of numerically solving an equation by means of 

 calculating sticks, with the only difference that the processes were 

 applied to the case of a literal quadratic equation, and were committed 

 to writing. This will be obvious when we make a comparison of the 

 two. In this matter the Japanese had advanced a step further than 

 the Chinese. The expansion of literal equations higher than the second 

 was executed in like manner. The circle-principle was effected ])y 

 applying such an expansion method and making the number of divisions 

 infinitely great. 



15. In the circle-measurement of early days, an arc was divided 



into 2, 4, 8, equal parts. But in the cubature of a sphere, 



it was divided into i)arts of equal height. For a spiral too, the arc 

 of the generating circle was divided into a number of equal parts. 

 These studies evidently had a fruitful influence on the treatment of 

 the circle. In his study of the circle-principle, Takebe Kenko ^p^^^i^ 

 (1664-1739) had early attempted an equal division of the chord instead of 

 the arc, but it remained for Ajima Naonobu ;^^ii[[il (P-1798) to achieve 

 success along this line, thus improving the circle-principle very conside- 

 rably. He effected the measurement of an arc of a circle by dividing its 

 chord into an infinite number of equal x)arts. His method may be compared 

 in some resj)ects with tlie Occidental method of definite integration. 



