摘要
"演段"是宋元数学中普遍使用的一个数学术语,也是宋元数学中最基本的代数方法,这一概念和方法伴随天元术在日本江户时代的传播和改造,成为和算中使用最为普遍的代数方法。然而,中日数学史学界对"演段"概念至今未能给出准确的解释。文章通过系统考察"演段"概念在中国和日本的流变以分析其内涵,认为:演是演算、推演的意思,段是"段数"的略称,所谓段数,指多项式的系数。演段,就是演算出多项式的系数,因此,它是以多项式为中心的代数演算方法。文章进一步论述了"演段"概念和方法在东亚的历史发展,认为它起源于汉唐时代列方程的几何图示法,在宋元时期逐渐由条段法转变为天元术,摆脱几何直观而成为以天元术为中心的代数演算方法,在明代随天元术的失传而转向衰亡。在江户时代的日本,由天元术发展出旁书法与消元法,丰富了"演段"的内涵,它包括天元术、旁书法(点窜术、天生法)与消元法(解伏题),但它作为以多项式为中心的代数演算方法的本质没有改变,它是东亚传统数学中的代数分析法,与西方笛卡尔的代数分析法东西辉映。
Yanduan（演段）was the primary algebraic method in the Song and Yuan dynasties and commonly used in mathematics of the period.This method was accompanied by the transmission and reconstructing of Tianyuanshu（天元术） in Japan＇s Edo（江户） period,and became the basic algebraic method in Wasan.However,the understanding of Yanduan was confused in China and Japan,and there is no exact interpretation as yet.This paper attempts to explore the meaning of Yanduan with a study on the course of its development in China and Japan,and concludes that Yan（演）denotes calculation and Duan（段） denotes the coefficient of polynomial,so Yanduan is an algebraic method of calculating the coefficient of polynomial with the polynomial as its center.The paper further discusses the historical development of Yanduan in East Asia,and states that it originated from graphic interpretation used in equation in the Han and Tang dynasties,and gradually evolved as Tianyuanshu from Tiaoduanfa（条段法）in the Song and Yuan dynasties,ending with the decline of Yanduan along with the loss of Tianyuanshu in the Ming dynasty.Nevertheless,the Hoshoho（旁书法）and elimination were evolved from Tianyuanshu in Japan in the Edo period and enriched the concept of Yanduan with TianyuanShu、Hoshoho and elimination.In addition,Yanduan is an algebraic analysis method in East Asian traditional mathematics and contrasts finely with the Descartes＇ methods in the West.
出处
《自然科学史研究》
CSCD
北大核心
2011年第3期318-344,共27页
Studies in The History of Natural Sciences
基金
国家社科基金重大招标项目"中外科学文化交流历史文献整理与研究"(项目编号:10&ZD063)
东华大学振兴文科计划预研究项目
关键词
演段
条段法
列方程
消元法
代数分析
Yanduan
Tiaoduanfa
equation
elimination
algebraic analysis
