割圆术近析

Modern Analysis of Division of a circle

本文对我国刘徽、日本关孝和及希腊海伦三位古代数学家在圆周率研究中的三种典型的割圆术作了分析比较.尤其是刘徽注中所说的“消息”一词作出了一种新的解释.对刘徽、关孝和的思想所定义的圆周长,以现代分析所常用的圆周长的定义为基础(即与海伦类似的方法)作了严格的证明.并对刘徽和关孝和的公式作了精度估计.文中指出了刘徽和关孝和的割圆术中提高精度的做法已具有本世纪初才提出的外推极限法思想.

This paper analyses and compares with the three typical sorts of division of a circle.They were made respectively by Chinese Liu Hui, Japanese Seki and Greek Heron. Particularly, We give a new explanation to the 'in formation' of liu Hiu's notes. By using modern anlysis (which is similar to Heron's method), the circumference formula defined by Liu Hui and Seki is strictly proved. The precision of their formula is estimated. It is remarkable , that the method used by Liu Hui and Seki to improve the precision, cortains the idea of extrapolation to the limit which was appeared anly in the beginning of this century.