190 YOSHIO MiKAMI 



Ajinia also succeeded in finding the volume of a circular cylinder 

 which is X)ierced perpendicularly by another, thus employing for the 

 first time the method of double integration. In this he made use of 

 a double series, which he integrated term by term and thus found the 

 resulting sum. The double or Uiultiple series had been used after 

 Kurushima's X"©^ f^i«^ Matsunaga's J^s^fa^p^ (.^-1744) times and the 

 expanded fornnilae for the })olygons had been obtained in the form of a 

 double series, but now Ajima was able to apply the double series to 

 problems of integration. After his days, the double or multiple series were 

 always abundantly used in the treatment of various complicated problems 

 of integration, which now came into existence. The integration method 

 of the circle-principle was again imi)roved by Wada fpfg .^ (1787-1840), 

 who consti'ucted a number of tables, by means of which the operations were 

 remarkably simplified. He also attempted to consider one of the parts 

 as equally divided any number of times, or the ^'differential" so to 

 speak, for the purpose of integration. Wada's construction of the 

 circle-principle tables was certainly made after the use had become 

 known of the logarithmic tables, which, imported direct from Holland or 

 studied through Chinese translations, were highly prized by Japanese 

 m athematicians . 



In the hands of the disciples of Ajima who iiad improved the 

 circle-principle, the rectification of an ellipse was effected by using the 

 double series. Wada again simplified the process and reduced the double 

 to a single series. At the same time, there arose a variety of problems, 

 which concerned the intersections of two solids, new curves and surfaces, 

 the sections of circular and elliptic wedges, etc But all these problems 

 concerned the quadrature, rectification and cubature of curves and 

 surfaces, whose properties had never before been discussed. The ellipse 

 was thoroughly studied, but the parabola received but slight attention, 

 while the hyperbola hardly came into notice except casually on the part 

 of careless problem -sol vers. Such a state of things happened because 

 the ellipse was commonly formed by cutting a cylinder and not a cone. 



16. Japanese mathematicians were accustomed to handle their 

 problems chiefly in an algebraical way. It was even so with problems 

 about geometrical figures. The term "tenzan problems," although 

 tenzan originally denoted nothing but algebra, came to be applied to 

 problems concerning geometrical figures, showing how algebraically 

 coloured Japanese mathematics had been. The cause is doubtless to be 

 traced to the Chinese superiority in algebraical matters and to the fact 



