【作者】 郭园园；

【导师】 纪志刚；

【作者基本信息】 上海交通大学 ， 物理学， 2013， 博士

【摘要】 本文是关于阿拉伯数学家阿尔·卡西（, al-Kāsh,约1380-1429年）所著《算术之钥》（Α Φ, Miftāh al-hisāb, The keyto Arithmetic，1427年）中代数学部分，即此书第五卷的研究。由于语言和史料等原因，至今国内外学者鲜有关于此内容全面和深入的研究。文中笔者首先详细解读《算术之钥》中的代数学内容，随后将其与早期阿拉伯数学著作及其它文明中的相关内容从算术化代数、高次开方法、“双试错法”和“百禽问题”四个主题进行研究。本文主要有三个方面的写作意义：首先，笔者在阿拉伯原典文献解读的基础之上，首次将《算术之钥》中代数学部分以及与其相关的早期阿拉伯数学著作中的内容全面系统地呈现出来；其次，本文对《算术之钥》代数学部分中四个主题的分析对于揭示此部分内容的来源及剖析阿拉伯中晚期代数学的发展脉络均有非常重要的意义；最后，笔者在高次开方法、“双试错法”和“百禽问题”的研究过程中，尝试性的进行了跨文明比较研究，这对于弥补传统相关研究中缺失的部分环节有积极作用。本文共分八章，各章要点如下：第1章绪论部分。介绍了阿拉伯数学史概况、本文选题背景及研究意义、阿拉伯数学史研究现状及本文研究进路。第2章在前人研究的基础上介绍了阿尔·卡西的生平及《算术之钥》的早期版本信息，此外根据搜集到的史料对近年来此书不同版本、研究文献、研究论文进行了汇总与梳理。第3章详细解读了《算术之钥》中代数学部分，即其第五卷的数学内容。第4章通过《算术之钥》与萨马瓦尔（，al-Samaw’al，约1130-约1180）《光辉代数》（Δ, al-Bahir of algebra，约1149）两书中算术化代数内容的比较研究，展现12至15世纪阿拉伯算术化代数的发展概况，可知这种源于方程化简过程中的基本运算步骤及简单的算术方法在阿拉伯数学家的努力下已经发展为一套完整的用于化简方程的理论体系。阿尔·卡西在《算术之钥》中对算术化代数在概念的表述、知识体系的理解以及具体算法方面都表现出与萨马瓦尔《光辉代数》中相关内容的相似性，可以判断阿尔·卡西书中的此部分内容延续了自12世纪以来的阿拉伯数学传统，但在具体的语言表述和算法方面又有明显的进步。第5章详细解析《算术之钥》中高次开方问题所涉及的一系列算法，并进行阿拉伯探源。此外将《算术之钥》中的高次开方内容与中算相关算法进行比较，可见二者在论述、方法等方面都有显著的不同。纵观阿拉伯数学史，对于这些不同点，卡西都保留着自10至12世纪以来的阿拉伯数学传统，可以判断《算术之钥》中的这些问题并没有像传统观点所认为的那样受到中国宋元数学的直接影响。第6章将《算术之钥》中“双试错法”（Φ Α Υ，Hisabal-Khata’ayn）与早期阿拉伯数学家寇斯塔·伊本·鲁伽（Γ,Qusta ibn Luqa,820-912）、萨马瓦尔和法雷西（, al-Farisi,约1260-约1320）著作中的类似算法从术文、证明等方面进行比较研究，得出阿尔·卡西的“双试错法”保留了自10世纪以来的阿拉伯数学传统，同时进一步剖析了“双试错法”在此时期阿拉伯代数学中作用和地位的演变；此外，还将阿拉伯“双试错法”与其它文明中的相似算法进行比较。第7章将《算术之钥》中的“百禽问题”与阿拉伯早期的阿布·卡米尔（Γ, Abu Kamil,约850-约930）、萨马瓦尔和法雷西著作中的相似问题从题目形式、问题本质及算法等方面进行了比较，得出阿尔·卡西与法雷西对“百禽问题”的认识可能是同源的。此外还研究了其它文明中的相关算题，从古代数学问题传播和演化的角度看，“百禽问题”与“双试错法”有明显的不同，这体现了古代数学文明在传播发展过程中的多样性特点。第8章结语更多还原

【Abstract】 The thesis is a research on the algebra in “The Key to Arithmetic”(Chapter V of thisbook) which was written in1427by Arabian mathematician al-Kāsh(1380-1429A.D.). Till now there have been seldom systematical researches on thisissue due to the obstacle of language and the lack of historical materials. This thesisinterprets the main contents of the algebra in “The Key to Arithmetic” in details,followed by a further comparative research on other correlative work in early Araband other civilizations from the following four aspects: the Arithmetization of Algebra,High-power Extraction,“Hisab al-Khata’ayn” and the “Problem of Hundred Birds”.The thesis has three significances. First, this thesis presents the contents of algebra in“The Key to Arithmetic” and some correlative Arab mathematical materialssystematically based on the interpretation of original Arabic text for the first time.Secondly, the research on the above four issues reveals the origin and the transition ofArabic algebra. Thirdly, this thesis makes comparative research tentatively betweendifferent civilizations on the last three issues, which have a positive effect on fillingthe gap of former researches.This thesis is composed of eight chapters, the points of each chapter as followed:Chapter I: Introduction. It includes the overview of the history of Arabic mathematics,the background and significances of the paper, the research situation of Arabicmathematics and research path of the paper.Chapter II: The detailed introduction of al-Kāsh, the information of the early versionsof “the Key to Arithmetic”, and the summary of various research literatures, papersand the versions based on the former research.Chapter III: The detailed interpretation of the algebra in “The Key to Arithmetic”,namely the mathematical part in Chapter5of the book.Chapter IV: This chapter shows the overview of “the Arithmatization of Algebra”during the12th-15thcentury by comparing the similar contents between al-Samaw‘al’s“al-Bahir” and “the Key to Arithmetic”, which shows that the basic steps which arederived from the process of equation simplification and simple arithmetic methodshave formed a complete theory system by the efforts of Arabian mathematicians. Itcan be conclude that the correlative content of “the Key to Arithmetic” has inherited the Arabic mathematical tradition from the12thcentury as the similarities between thetwo books on three aspects: the presentation of concepts, the understanding ofknowledge system and the specific algorithms. Nevertheless，al-Kāsh has madeobvious progress on the aspects of the representation and the algorithms.Chapter V: Based on the interpretation of original Arab text, this chaptersystematically analyses the operation of the high-power extraction and a series ofalgorithms in “The Key to Arithmetic” and trace to its Arab origin. In addition thischapter compares the correlative contents in “The Key to Arithmetic” with Chinesetraditional mathematics, which shows the obvious differences between them from theperspective of the text and method. Throughout the history of Arabic mathematics,most of the algorithms in “The Key to Arithmetic” have inherited the tradition ofArabic mathematics during the10thto the12th, from which we can concluded that thealgorithms in “the Key to Arithmetic” are probably not be influenced directly by themathematics of Song and Yuan Dynasties in China, as the former researchers haveaffirmed.Chapter VI: This chapter compares the “Hisab al-Khata’ayn” with “the Key toArithmetic” and other early Arabian mathematical works, such as Qusta ibn Luqa(820-912A.D.), al-Samaw’al and al-Farisi (1260-1320A.D.) from the aspects of thetexts and proofs, which comes to the conclusion that, the “Hisab al-Khata’ayn” in “theKey to Arithmetic” has inherited the tradition of Arabic mathematics from the10thcentury. This chapter also analyzes the tradition of “Hisab al-Khata’ayn” in Arabicalgebra, comparing it with the similar algorithm in the medieval Europe mathematicsand Chinese traditional mathematics.Chapter VII: This chapter compares “the problem of Hundred Birds” in “the Key toArithmetic” with the similar problem in the books of other early Arabianmathematicians, such as Abu Kamil (850-930A.D.), al-Samaw’al and al-Farisi, andthe mathematicians in other civilizations, such as Leonardo Fibonaci (1170-1250A.D), Zhang Qiujian (5thcentury) and Zhen Luan(535-566A.D.) from the aspectsof the problem forms, the essence of problems and the algorithm. Although there is nohistorical materials to prove the relationship of the problem between al-Farisi andal-Kāshī’s work, it can be concluded that they may derive from the same root. Inaddition, this chapter also discusses the correlative problems in other civilizations,with conclusion that “the problem of Hundreds Birds” is quite different from “theHisab al-Khata’ayn” which shows the diversity of the ancient mathematics during itsspread across the cultures.Chapter VIII: Epilogue.更多还原