194 YOSHIO MiKAMI 



of which solutions were obtained by wliolo numbers. Those j»roblems 

 in which a ci'own of successively circumscribed circles was considered, 

 come under our present heading, thus showing an intimate relation 

 with the progress of treating successively described circles. Of this kind 

 of problem, some vei'y beautiful solutions were obtained by Ajima 

 and Gokai ^^^^- Japanese mathematicians solved in integers not 

 only problems al>out certain specified geometrical figures, but used also 

 always to give integral value solutions for all the sorts of geometrical 

 ])i-oblems which they considered. 



The calculations of the integers for a right triangle were soon 

 extended to the case of a rectangular trapezoid. A.t the same time, 

 there were considered a number of indefinite equations without making 

 any reference to geometrical figures. Most meritorious of these were 

 the equations 



y2= X? + Xi + , + xf, 



f = Ixf + 2x1 + 3x? + + nxl 



y" = Ixj + 3x| + 6x3 + + Un xf, , etc. 



x^ + 3'^ = z^ + u"'. 

 The factorization of numbers and the section of prime numliers were 

 also attempted, leading to worthy results. 



The]-e were numerous studies in connection with magic squares, 

 which had been first imported from China. The ancient Chinese diagram 

 called Lo-shu '^^ was a contrivance for arranging the first nine natural 

 numbers in a magic square, certainly the oldest example of the kind 

 in the world. Other magic squares and magic circles were recorded in 

 a work of the 13th century and were reproduced in the Suan-fa T^img- 

 tsung ^^^^^ of 1593. It was these that had stimulated the Jai)anese 

 to make further studies along the same line. In Japan many interesting 

 results were obtained that were not encountered in China. Japanese 

 mathematicians also attempted to find out the greatest possible number 

 of magic squares which could be arranged with certain given numbers. 

 Magic cubes were noticed too, although very rarely. 



18. So far, we have given a rough general survey of the work of 

 the old Japanese mathematians. It is of course out of the question 

 for us to give every detail of their work in this short article. The 

 development of differentiation, of the theory of limits, and of the 

 treatment of continued fractions, as well as the cubature of certain 

 solids of revolution and numerous other themes, have all been omitted. 

 Japanese mathematicians learned very much at the outset from 



