188 YOSHIO Mi KAMI 



for certain types of proLlenis, and thus there appeared a number of 

 methods of successiv^e approximations. There were tried also several 

 ways of expanding the root of an equation in the form of an infiidte 

 series or other infinite expressions. The Knisliiki Shimpo l|^^0f'/i 

 of 1803 vvas one work dealing with this type of problem. The treatise 

 elucidates a process of approximating b}^ degrees all the real roots of 

 a rational algebraical equation of any degi-ee. 



13. The kahujutsu ^^"jtj, or ''polygonal theory," relates to the 

 measurement of regular polygons. After Seki, attempts were made to 

 construct equations representing the radii of the circumscribed and 

 inscribed circles of a regular polygon in terms of one of its sides. Seki 

 gave results from the triangle up to the 20-gon. Soon mathematicians 

 began an examination of these equations in order to obtain the formulae 

 in a general form, the considerations being carried out inductively. 

 A method called sTiokaku totetsu-jntm ^f\WiW\M ^^^^ based on this 

 way of induction. Accompanying the development of the circle-measure- 

 ment, the polygonal theory underwent a related formulation and there 

 appeared some formulae expressed in double series. Among those 

 engaged in this study, Kurushima Gita ^^^^ii^^i^-Vlbl) did some 

 good work. There are a large number of documents about this theory, 

 nearly all of them being manuscripts. 



Inductive considerations were profusely used by Jai)anese mathema- 

 ticians, being applied to a variety of problems, but the inductions 

 employed were all imperfect. Seki and others used the method of 

 induction for determinants though imperfectly and also for the 

 enumeration of piles of various kinds, by which process a large number 

 of interesting formulae were obtained. Jlenjatsit ^|)t5 and cliikusaku-jutsu 

 ^^ir^ were mathematical methods that related to the application of 

 inductive reasoning to obtain successive values for a certain quantity. 

 The polygonal theory was one of the problems to which such a method 

 had been applied. It appears also that tlie method of mathematical 

 induction was nearly arrived at. 



14. Circle-measurement was also early studied by Japanese 

 mathematicians. The method given in the KioaUuyo Sampd tS^Wfi 

 of 1709 may be noticed first of all. Similar considerations appeared 

 in various other works. Within a circular arc were inscribed two equal 

 chords, and .the nund);?r of inscribed chords was successively doubled ; by 

 this means the values of the successively inscribed perimeters or their 

 squares were calculated, a kind of treatment that is reminiscent of the 



