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中文题目: 杨辉《详解九章算法》(1261 CE)方程章算法与题目顺序初探
英文题目: Preliminary Research on the Mathematical Methods and Order of Problems in the “Fangcheng” Chapter of Yang Hui’s Mathematical Methods (1261 CE)
作 者: 周霄汉    
刊物名称: Chinese Annals of History of Science and Technology
发表年度: 2020
卷: 4
期: 1
页码: 1–62
中文摘要:

杨辉是中国13世纪数学著作的重要作者之一,《详解九章算法》(1261 CE)是现存杨辉最早的一部著作。1315世纪,杨辉的这部著作在《九章算术》的流传与流行中起到了关键的作用。然而,现存的《详解》刻本残缺不全,并且包含大量舛误,这些阻碍了当代研究者对这部著作的理解。《九章算术》的方程章处理有关我们现在所视作线性方程组的问题,然而阐释《详解》中该章的文本并且恢复与方程有关的数学操作却有困难。通过细致的文本和算法分析,本文作者解释了杨辉对于方程法正负法的理解和操作。《详解》中与正负法相关的一些文字含义模糊,本文的附录是对这些文字的细致的翻译,并且给出了支持这种阐释的理由。杨辉对于概念的理解以及他的与这两个概念相关的操作,很容易与现代数学中的相应内容相混淆。对这部著作中算法的仔细分析揭示了杨辉《纂类》(即《详解》最后一部分)中问题的顺序是依据《详解》中算法的注释而安排的。《详解》注释中的一些文本线索提到前问,实际上反映的是《纂类》中的题目顺序。杨辉重新安排了一些题目的顺序,使之不同于古代经典《九章》中原来的顺序,并对这些改变顺序的题目做了详细的注释。所有这些发现揭示并证实了《详解》和《纂类》之间的密切关系。

英文摘要:

Yang Hui was one of the most important authors of mathematical works during the thirteenth century. Mathematical Methods Explaining in Detail The Nine Chapters (Xiangjie jiuzhang suanfa 詳解九章算法, 1261 CE) is the earliest extant work attributed to Yang Hui. From the thirteenth to the fifteenth century, this work played a crucial role in the circulation and popularization of The Nine Chapters on Mathematical Procedures (Jiuzhang suanshu 九章算術). However, the only surviving printed edition of Mathematical Methods is incomplete and contains many mistakes obstructing contemporary researchers’ understanding of this work. The “Fangcheng” chapter of The Nine Chapters deals with problems related to solving what today are known as simultaneous sets of linear equations. However, interpreting the text in this chapter of Mathematical Methods and recovering the mathematical practices relating to fangcheng are difficult. Through detailed textual and mathematical analyses, the author of this paper explains Yang Hui’s understanding and practice relating to “the fangcheng method” and “the method of the positive and the negative.” This paper includes an appendix that provides a detailed translation of the ambiguous text relating to “the method of the positive and the negative” and gives reasons supporting the interpretation provided here. Yang Hui’s understanding of the concepts of “positive” and “negative” and his practice relating to these two concepts may easily be confused with their apparent counterparts in modern mathematics. Also, careful analysis of the mathematical methods in this work reveal that the order of problems in Yang Hui’s Reclassifications of Mathematical Methods Explaining in Detail The Nine Chapters ([Xiangjie jiuzhang suanfa zuanlei 詳解九章算法纂類], namely, the last section of Mathematical Methods) were rearranged according to commentaries to specific methods that appear in Mathematical Methods. Some textual clues referring to the “previous question” (qianwen 前問) in certain commentaries of Mathematical Methods indeed reflect the order of problems in Reclassifications. Yang Hui made especially detailed commentaries on the problems that he arranged in a sequence that differs with respect to the original order of problems as they appear in the ancient classic work, The Nine Chapters. All these discoveries reveal and serve to prove a close relationship between Yang Hui’s Mathematical Methods and his Reclassifications.

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