On the Missing Note of T Seki:"Kongenki Enjutsu 16 Problems"

Abstract

There used to be a field called Senkyo problem or Common Part Problem in Wasan which the old Japanese mathematicians or Wasan experts earnestly studied during the Edo period. We have already explained in some journals that Takakazu Seki (1642?-1708) was able to solve the problems without using integral calculus. This time, we have found a new description about the missing note of T Seki in the introductory remarks and in the main body of KTangen Sanpo", which was wr ten by Shukei Irie in 1739 According to Irie's description, he called it uKongenki Enjutsu 16 Problems" And in the main body of the text Irie had cited, in order to solve a Senkyo Problem, that T.Seki had used an approximate formula to find the area of a segment of a circle. We were able to restore this approximate formula as follows: If we let d be the chord, c the altitude of a segment of a circle, and S the area, we have, (2d+c)cπ/10=S Through research of Irie's statement, regardless of it being true or not, we obtained some new facts about T. Seki as follows: Firstly, it is obvious that Seki studied "Sanpo Kongenki" written by Seiko Sato in 1669, from which he learned an approximate formula like the one mentioned above. We believe that this matter may create a new point of view on the study of T. Seki. Secondly, T. Seki must have made a note called "Kongenki Enjutsu 16 Problems" immediately after "Kokon Sanpoki" by Kazuyuki Sawaguchi was published. This is because K. Sawaguchi did not solve 16 out of 150 problems in "Sanpo Kongenki", which Sato poured out as new questions for Wasan experts of that time. Thus, we are able to place the missing note in an early time of his work. Thirdly, it is certain that Seki's successors have passed on this missing note for mathematical education and it existed until around the end of the first half of the 18th century.