192 YnSHIO MlKAMI 



Tlie prol)]em ])rol)aLly inost worthy of notice is the one projjosed 

 by Sawaguchi f|lp— ;^ (1670) and solved by Seki, in which a great circle 

 was circnm scribed around a middle circle and two eqnal small circles. 

 Various modifications occurred. Sometimes the two small circles were 

 made nnequal. Then another circle was added also in contact. Lastly 

 there were inscribed successive circles always in contact. Similar figures 

 were also considered where the successive circles were described touching 

 two intersecting circles, and the like, cases where the successively 



described circles form a crown also being considered. There was 

 devised a method called chihisaku-jidm ^^^\1^ for the evaluation 

 of the diauieters of these successively inscribed circles. The method 

 evidently bears a close relation to the recurring calculations of 

 successive values, a]»plied by Japanese mathematicians to various sorts 

 of problems. It i»roved a long step forward in tlie advancement of 

 Jai)anese geometry. 



Another important problem was that which related to three circles 

 inscribed within a triangle. This problem obviously corresponds to 

 Malfatti's problem ; and it Avas solved before Mali\itti. It was not 

 considered, however, as a problem, of construction, but as one of 

 evaluating the diameters of the inscribed circles. This problem was 

 soon extended to the case of inscribing four circles. There were also 

 considered prolilcms in which circles were inscribed between three circular 

 arcs. And further there ap])eared a number of figures, in which successive 

 circles were variously inscribed, this problem being solved by the 

 application of the above mentioned process of successive calculations. 

 Ajima tried to make use of a geonietrical relation concerning the 

 tangent common to two circles for the solution of a large number of 

 problems of contact. Some years later there appeared another relation 

 for the same, which was more convenient in application. 



