戴煦对欧拉数的研究

DAI XU'S RESEARCH ON EULERS NUMBERS

戴煦(1805—1860)的《外切密率》出版于1852年。在这部著作中,他继承了明安图(约1692—1763)、董祐诚(1791—1823)、项名达(1789—1850)等人在三角函数的无穷级数展开式上的工作,并有所创新。当时数学家们探讨了正弦、余弦、正矢、余矢的幂级数,而对正切、余切、正割、余割的幂级数尚未涉及。项名达曾注意过此问题,但未能解决,"以弧分不通切割为憾"。戴煦经多年钻研,终于得到了这一问题的答案,著《外切密率》,"推衍数术、以呈先生(项名达),而先生以未有术解为嫌",其后又作了修改补充。1851年,戴煦结识李善兰,李氏非常欣赏《外切密率》,遂互相研讨、切磋琢磨,经再三敦促,戴煦终于将此书定稿出版。

Dai Xu (1805-1860), the Chinese mathematician of the Qing Dynasty, in Chapter 2 of his Wai Qie Mi Lu (On the Determination of Segment Areas, 1852) introduced independently Euler numbers E_n for the expansion to sec a. He employed a recursive method and put forward several counting functions, one of which is equivalent to a recurrence formula E_n: He got E_7= 199,360,981. E_8= 19,391,512,145. E_9=2,404,879,675,441.This paper deals with Dai Xu's methods and results. Its emphasis is on the analysis of the text and the elucidation of his methods. We wish to show that although 19th-century Chinese mathematics had fallen behind the West there still remains something worthy of our attention today and can serve as a staring point in modern research.