Infix Operation Count By Study

Takebe Katahiro (建部贤弘 1664—1739) was generally called tolajiro (彦次郞) named Fukyu (不休), and he was born of gentry in Edo (now Tokyo). He was an officer of Tokugawa Bakuhu (德川幕府) from 1692 to1733, once being an astronomical counselor of the eighth Tokugawa Shougun,Yoshimune(八代将军德川吉宗, reign from 1716 to 1745) and surveying and mapping the map of all over Japan with orders. In 1676, when he was 13 years old, he became a student of Seki Takakazu (关孝�'�, 1642? —1708) and began to learn mathematics. Afterward he became the most outstanding and creationary Wasan mathematician after Seki Takakazu. Tetsujutsu Sankei ( 《缀术算经》 ,1722), his magnum opus of mathematics, was the most important classic in the history of Wasan, so is it in the history of mathematics in the Chinese-Character Cultural Circle. All the creative results of mathematics made by Takebe Katahiro are contained in Tetsujutsu Sankei, such as: first, as for the calculation of number π, he created the "Repeated Shoyakujutsu Method" which is a kind of the numerical approaching method, getting the number of π with exactness till the 41^st place after decimal point depending on the method. Then, moreover, he calculated the length of arc with the method, inaugurating the foundation of studying on infinite series in Wasan. At last, he used the method of derivative to calculate the extremes of polynomial function for the first time in the history of Wasan, and afterwards Wasan approached to the age of infinitesimal analysis. All the achievements mentioned above represented a peak of studying of mathematics in Japan at that time, and, at the same time, these were also a turning point in studying of Enri Algorithm (圆理,Circle Theory). The most important thing was that the book(Tetsujutsu Sankei) was the first one to discuss the mathematical thought and mathematical methodology, in which Takebe tried to build the general method called Tetsujutsu (缀术) basing on induction.In the paper, the author studies the Tetsujutsu Sankei systemically under the view of the Chinese-Character Cultural Circle, even of the traditional mathematics in China.Firstly, reviewing the social and scientific contexts during the Edo period when Takebe Katahiro lived, by which the author tries to give the reason rationally why Takebe Katahiro made the eminent achievements, gives the textual research of his life and summarizes his mathematical achievements.Secondly, depending on the textual research of the time of each hand-copied book accomplished , and the discussing of the main differences among the three editions, the author makes the conclusion that Tetsujutsu Sankei(National Archives of Japan,内阁文库本) which is Takebe Katahiro 's manuscript is the most early, then is Fukyu Takebe-Sensei Tetsujuteu(University of Tokyo, 东大本) adapted by later generations, and...