《九章算术》环田题“密率术”考辩

A Textual Criticism on the "Art of Mi-Lü" about the Ring Measurement in the "Nine Chapters on Arithmetics"

“密率术”是《九章算术》校勘中,迄今尚未解决的一大疑案。清代著名学者戴震,删去“密率”二字而将术文校改为环田面积的两种算法;后世学人大多因袭其说。本文则指出戴氏校订之谬失,辩明“密率术”乃依据环田周径之数推求圆弧与直径之比率的算法。

In collating the 'Nine Chapters on Arithmetics' researchers have left the problem of the 'Art of Mi-Lu' still unsettled. Dai Zhen (AD.1723～1777), famous scholar of the Qing Dynasty, deleted the term 'Mi-Lu' and turned the original into two methods of calculating the area of a ring; most scholars in later ages followed his explanation as a final conclusion without any dissent. The present article is to point out Dai's error and provide an explanation of the 'Art of Mi-Lu as an art of calculating the ratio of the circumference or the arc of a circle to its diameter according to the inner and outer circumferences and the width of a ring.