《算术之钥》之代数学研究
|
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
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.
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
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.
引文
1（美）Berggren, J. Lennart,"Mathematics in Medieval Islam", in Victor J. Katz, The Mathematics of Egypt,Mesopotamia, China, India, and Islam: A Sourcebook (Second ed.), Princeton, New Jersey: Princeton University,2007.
2（英）F.Robinson.Cambridge Illustrated History of Islmic World[M].北京:世界知识出版社.2005. P299-302.
3（美）霍华德伊夫斯（著），欧阳绛（译），数学史概论（第六版）[M].哈尔滨：哈尔滨工业大学出版社，2009. P237-238.
5Joseph, George Gheverghese, The Crest of the Peacock: Non-European roots of Mathematics, Harmondsworth:Penguin,1992.P344.
6（美）Katz, Victor J., A History of Mathematics: an Introduction, Reading MA: Addison-Wesley,1998.P240.
7吴文俊,中国数学史的新研究（王志健译）.自然杂志，1989,12（7），P.546-550.
8吴文俊,关于研究数学在中国的历史与现状--《东方数学典籍<九章算术>及其刘徽注研究》序言,自然辩证法通讯,1990,68（1），P37-39.
9李文林，丝绸之路数学名著译丛（导言），北京：科学出版社，2008.Pxi-xii.
10（英）F.Rosen (trs.), Muhammad ibn Musa Al-Khwarizmi: Algebra.London,1831.
11Jamshid al-Kāshī. Miftah al-Hisab (Key to Arithmetic)[M]. A S al-Demerdash.M H al-Cheikh (eds). Cairo: D al--,1967.
12Al-Khwarzmi’s Algebra[M], A Reprint of the Rosen s Translation of1831,Pakistan HijraCouncil,Islamlabad,1989,No.80of the Great Books Project.
13杨淑辉,张妍.数学史家尤什克维奇的杰出贡献[J].辽宁师范大学学报(自然科学版),2010,2(4):161-164.
15Al-，B.A.Rosenfeld（译）,Rosenfeld&A.P.Youschkevitch（评注,Mathematicheskie traktaty，Istorikomatematicheskie issledovaniya Volumu VII，1954.其中《算术之钥》的译文位于PP.13-326.
16Al-，B.A.Rosenfeld（译）,V.S.Segal&A.P.Youschkevitch（编辑）,Rosenfeld和A.P.Youschkevitch（评注），Klyuch Arifemetiki; Traktat ob Okruzhnosti (The Key to Arithmetic; The Treatise on the Circumference)，Moscow：Gosudarstvennoe Izdatel stvo Tekhniko-teoreticheskoi Literatury，1956.其中《算术之钥》的译文位于PP.7-262，《论圆周》的译文位于PP.263-319，随后是评注，最后于PP.428-568附有《算术之钥》莱顿手稿的照片。
17R Rashed.The Development of Arabic Mathematics: Between Arithmetic and Algebra[M]. A F W Armstrong(trans). Kluwer Academic Publishers,1994.p.201, note8.
19Jamshid al-Kāshī, al-Nabulsi,Miftah al-Hisab (Key to Arithmetic), Damascus, Syris: Ministry of High Education,1977.
20（德）Paul Luckey，Die Rechenkunst bei Gamsid b.Masud al-Kasi，Wiesbaden: F, Steiner,1951.
21Roshdi.Rashed, Entre arithmetique at algebra: recherché sur I’histoire des mathematiques arabes, Paris: lesBelles letters,1984.
22（法）Roshdi Rashed, trans, by A.F.W.Armstrong, The development of Arabic mathematics: between arithmeticand algebra, Dordrecht: Boston; London: Kluwer academic,1994.
23P.Luckey, Die Ausziehung des n-ten Wurzel und der binomische Lehrsatz in der islamischen Mathemarik, Math,Annalen120,1948, pp.217-274.
24（美）Abdul-Kader Dakhel, Edited by Wasfī A.Hijab and E.S.Kemmedy, The Extraction of the n-th root in thesexagesimal notation, a study of chapter5, treatise III of by Jamshīd Ghiyāth al-Dīn al-Kāshī with translationand commentatry, Beirut: American University,1960.
25R Rashed.The Development of Arabic Mathematics: Between Arithmetic and Algebra[M]. A F W Armstrong
(trans). Kluwer Academic Publishers,1994.P143,Note,10.
26萨马瓦尔的主要著作有两本：一本是《光辉代数》1972年由S.Ahmd和R.Rashed共同校订出版，其为阿拉伯文版并附有法文简介（S.Ahmd and R.Rashed eds., al-Samaw al, al-Bahir en algebra, Damascus,1972）；另外一本是名为al-Qiwami fi al-Hisab al-Hindi(1172-1173)的算术著作，此书全文已经遗失，但是其第三卷的手稿得以保存，详见R Rashed.The Development of Arabic Mathematics: Between Arithmetic and Algebra[M]. AFW Armstrong (trans). Kluwer Academic Publishers,1994.p143, Note8.
27P Luckey, Die Rechnenkunst bei Gamsid b. Masud al-Kasi (Wiesbaden,1951).
28R Rashed, The development of Arabic mathematics: between arithmetic and algebra (London,1994).P127-134.
30B A Rosenfeld, A P Youschkevitch, Biography in Dictionary of Scientific Biography (New York1970-1990).P256.
31E.S.Kennedy, The Planetary Equatorium of Jamshid Ghiyath al-Din al-Kāshī[M].Princeton Oriental StudiesVolume18.Princeton University Press,1960,P1.
33B A Rosenfeld, A P Youschkevitch, Biography in Dictionary of Scientific Biography (New York1970-1990). P256.
34M Bagheri, A newly found letter of al-on scientific life in Samarkand, Historia Math.24(3)(1997),241-256.
35P.Luckey,Der Lehrbrief Uber den Kreisumfang von Gamsid b. Masud al-Kasi, Klasse fur Mathematik undallgemeine Naturwissenschaften Jahrgang,1950,Nr.6.
36B.A.Rosenfeld, A P Youschkevitch, Biography in Dictionary of Scientific Biography (New York1970-1990). P257.
37赵栓林.对《代数学》和《代数术》术语翻译的研究[硕士论文].2006:内蒙古师范大学.
38艾约瑟.阿尔热巴喇附考,中西闻见录第七号.1873.
39（美）卡尔B鲍耶（CARL.B.BOYER著）,（美）尤塔C梅兹巴赫（UTA.C.MERABACH修订）,秦传安（译）.数学史（上册）[M].北京:中央编译出版社.2012.P256.
50R.Rashed et. B.Vahabzaded. Al-Khayyam Mathematicien[M]. Paris: Librairie Scientifique et Technique AlbertBlanchard,1999. P125-129.
51S.Ahmd and R.Rashed eds., al-Samaw al, al-Bahir en algebra, Damascus,1972.P119.
52（意）斐波那契原著，（美）西格尔英译，纪志刚、汪晓勤、马丁玲、郑方磊译.计算之书.北京:科学出版社，2007.P513.
53Jamshid al-Kāshī. Miftah al-Hisab (Key to Arithmetic)[M]. A S al-Demerdash.M H al-Cheikh (eds). Cairo: D al--,1967.P211.
54S.Ahmd and R.Rashed eds., al-Samaw al, al-Bahir en algebra[M], Damascus,1972.P140.
61Jamshid al-Kāshī. Miftah al-Hisab (Key to Arithmetic)[M]. A S al-Demerdash.M H al-Cheikh (eds). Cairo: D al--,1967. P225.
69Jamshid al-Kāshī. Miftah al-Hisab (Key to Arithmetic)[M]. A S al-Demerdash.M H al-Cheikh (eds). Cairo: D al--,1967.P234.
82T.L.Heath, The Thirteen Books of Euclid s Elements[M], New York: Dover Publications, INC.1956. Volume I, P71.
83Jamshid al-Kāshī. Miftah al-Hisab (Key to Arithmetic)[M]. A S al-Demerdash.M H al-Cheikh (eds). Cairo: D al--,1967.P264.
87Jamshid al-Kāshī. Miftah al-Hisab (Key to Arithmetic)[M]. A S al-Demerdash.M H al-Cheikh (eds). Cairo: D al-al-,1967.P268.
137S.Ahmd and R.Rashed eds., al-Samaw al, al-Bahir en algebra, Damascus,1972.P63.
159Jamshid al-Kāshī. Miftah al-Hisab (Key to Arithmetic)[M]. A S al-Demerdash.M H al-Cheikh (eds). Cairo: D1a6l-0-,1967.P66-67.
163R Rashed.The Development of Arabic Mathematics: Between Arithmetic and Algebra[M]. A F W Armstrong
(trans). Kluwer Academic Publishers,1994.P109-110.
165R Rashed. Combinatorial analysis, numerical analysis, Diophantine analysis and number theory[A]. R Rashed:Encyclopedia of the History of Arabic Science[C]. New York: Routledge,1996.P383.
170Jamshid al-Kāshī. Miftah al-Hisab (Key to Arithmetic)[M]. A S al-Demerdash.M H al-Cheikh (eds). Cairo: D al--,1967.P73.
177R Rashed.The Development of Arabic Mathematics: Between Arithmetic and Algebra[M]. A F W Armstrong(trans). Kluwer Academic Publishers,1994.P136.
178Al-Samaw al. al-Bahir en algebra[M]. S Ahmd.R Rashed (eds). Damascus:Imp. De I Universite deDamas,1972.P111.
179Jamshid al-Kāshī. Miftah al-Hisab (Key to Arithmetic)[M]. A S al-Demerdash.M H al-Cheikh (eds). Cairo: D al--,1967. P73.
182P Luckey. Die Ausziehung des n-ten Wurzel und der binomische Lehrsatz in der islamischen Mathemarik[A].Springer:Math Annalen120,1948.217-274.
183杜石然.试论宋元时期中国和伊斯兰国家间的数学交流[A].见:钱宝琮.宋元数学史论文集[C].北京:科学出版社,1966.P258.
184杜石然.再论中国与阿拉伯国家间的数学交流[J].自然科学史研究,1984,3(4):294-303.
185钱宝琮.增乘开方发的历史发展[A].见:钱宝琮.宋元数学史论文集[C].北京:科学出版社,1966.36-44.
186郭书春.九章算术译注[M].上海:上海古籍出版社,2007.133.
188李继闵.刘徽关于无理数的论述[J].西北大学学报（自然科学版）.1989,19(1),P1-4.
189钱宝琮.增乘开方发的历史发展[A].见:钱宝琮.宋元数学史论文集[C].北京:科学出版社,1966.P37-38.
190杜石然.试论宋元时期中国和伊斯兰国家间的数学交流[A].见:钱宝琮.宋元数学史论文集[C].北京:科学出版社,1966.P259.
194S.Ahmd and R.Rashed eds., al-Samaw al, al-Bahir en algebra, Damascus,1972.P160.
208（意）斐波那契原著，（美）西格尔英译，纪志刚、汪晓勤、马丁玲、郑方磊译.计算之书[M].北京：科学出版社，2007.P526.
214郑方磊.许凯《算术三编》研究.上海交通大学硕士论文.2007.p47.
215马丁玲.斐波那契《计算之书》研究.上海交通大学博士论文.2009.p128.
216钱宝琮.《九章算术》盈不足术流传欧洲考.《钱宝琮科学史论文集》.北京：科学出版社，1983.P83-96.
217杜石然.试论宋元时期中国和伊斯兰国家间的数学交流《.宋元数学史论文集》.科学出版社,1966.P241-265.
218郭书春，刘钝校点.《算经十书》（一）.沈阳：辽宁教育出版社，1998.
219（明）李之藻（演）,（意）利玛窦（授）.同文算指.郭书春.中国科学技术史典籍通汇.数学卷.第4册.郑
222Sesiano, Jacques, Abū Kāmil, Encyclopedia of the history of science, technology and medicine innon-weatern cultures. Springer1997-07-31. P4-5.
223Sesiano, Jacques, An introduction to the history of algebra: solving equations from Mesopotamian times toRenaissance. AMS Bookstore,2009. P75-76.
224Helaine Selin(ed.), Mathematics Across Cultures: The History of Non-Western Mathematics. Kluwer Academic
235Jamshid al-Kāshī. Miftah al-Hisab (Key to Arithmetic)[M]. A S al-Demerdash.M H al-Cheikh (eds). Cairo: D al--,1967.P243.
242朱一文.百鸡术的历史研究.上海交通大学硕士论文.2008.P29-30.
243张丘建（南北朝）.张丘建算经[A].中国科学技术典籍通汇数学卷[Z].第1册.郑州：河南教育出版社，1993.
244甄鸾（南北朝）.数术记遗[A].郭书春，刘钝.算经十书[Z].沈阳：辽宁教育出版社，1998.
245（意）斐波那契原著，（美）西格尔英译，纪志刚、汪晓勤、马丁玲、郑方磊译.计算之书.北京:科学出版社，2007.P261-262.
248（意）斐波那契原著，（美）西格尔英译，纪志刚、汪晓勤、马丁玲、郑方磊译.计算之书.北京:科学出版社，2007.P265-266.
259Al-Khwārizmī,Edited with translation and commentary by Roshdi Rashed, The Beginnings of Algebra[M], SAQI,Landon,2009.
260Berggren, J. Lennart,"Mathematics in Medieval Islam", in Victor J. Katz, The Mathematics of Egypt,Mesopotamia, China, India, and Islam[M]: A Sourcebook (Second ed.)[M], Princeton, New Jersey: PrincetonUniversity,2007.
261J.L. Berggren，Episodes in the mathematics of medieval Islam. Springer,2003.
262Jamshid al-Kāshī, Miftah al-Hisab (Key to Arithmetic)[M]. A S al-Demerdash.M H al-Cheikh (eds).Cairo: Dār al-kātib al-arabī,1967.
263郭园园，《算术之钥》与中算若干问题的比较研究[J].自然科学史研究,2012,31(1),P107-128.
[1] Al-Khwārizmī,Edited with translation and commentary by Roshdi Rashed, TheBeginnings of Algebra[M], SAQI, Landon,2009.
[2] F.Woepcke, E\XVa X Y FaO V, XVa X 'aP bV TaV AbSY B OV MS aQQ B RAlhacan Alkarkhi[M], Reprint of the1853original (Hildesheim,1982).
[3] S.Ahmd and R.Rashed eds., al-Samaw’al, al-Bahir en algebra[M], Damascus,1972.
[4] Jamshid al-Kāshī, Miftah al-Hisab (Key to Arithmetic)[M]. A S al-Demerdash.MH al-Cheikh (eds). Cairo: Dār al-kātib al-arabī,1967.
[5] Jamshid al-Kāshī,al-Nabulsi,Miftah al-Hisab (Key to Arithmetic)[M], Damascus,Syris: Ministry of High Education,1977.
[6] R.Rashed et. B.Vahabzaded, Al-Khayyam Mathematicien[M]. Paris: LibrairieScientifique et Technique Albert Blanchard,1999.
[7] Jacques Sesiano, Books IV to VII of Diophantus Arithmetica in the Arabictranslation attributed to Quatā ibn Lūqā[M], Sources in the History of Mathematicsand physical Sciences3. Springer-Verlag New York, Heidelberg, Berlin.1982.
[8] Kamal al-Din al-Farisi, Asas al-qawa ld fi usul al-fawa ld (The base of the rules inthe principles of uses)[M], edited by M.Mawaldi, Cairo: The Institute if ArabicManuscripts,1994.
[9] L.E.SIGLER, Fibonacci s Liber Abaci (A Translation into Modern English ofLeonardo Pisano’s Book of Calculation)[M]. Springer,2003.
[10]（意）斐波那契原著,（美）西格尔英译,纪志刚,汪晓勤,马丁玲,郑方磊译.计算之书[M].北京:科学出版社,2007.
[11](印)婆什伽罗原著,（日）林隆夫译注,徐泽林,周畅,张建伟译,莉拉沃蒂[M].北京:科学技术出版社,2006.
[12]张丘建（南北朝）,张丘建算经[A].中国科学技术典籍通汇数学卷[Z].第1册.郑州：河南教育出版社，1993.
[13]甄鸾（南北朝）,数术记遗[A].郭书春,刘钝.算经十书[Z].沈阳：辽宁教育出版社，1998.
[14]（明）李之藻,（意）利玛窦[A].同文算指.郭书春.中国科学技术史典籍通汇.数学卷[Z].第4册.郑州：河南教育出版社,1993.
[1] Berggren, J. Lennart,"Mathematics in Medieval Islam", in Victor J. Katz, TheMathematics of Egypt, Mesopotamia, China, India, and Islam[M]: A Sourcebook(Second ed.)[M], Princeton, New Jersey: Princeton University,2007.
[2] P.Luckey, Der Lehrbrief Uber den Kreisumfang von Gamsid b. Masud al-Kasi,Klasse fur Mathematik und allgemeine Naturwissenschaften Jahrgang[M],1950,Nr.6.
[3] Joseph, George Gheverghese, The Crest of the Peacock: Non-European roots ofMathematics[M], Harmondsworth: Penguin,1992.
[4] Katz, Victor J., A History of Mathematics: an Introduction[M], Reading MA:Addison-Wesley,1998.
[5] B A Rosenfeld, A P Youschkevitch, Biography in Dictionary of ScientificBiography [M](New York1970-1990).
[6] E.S.Kennedy, The Planetary Equatorium of Jamshid Ghiyath al-Din al-Kāshī[M].Princeton Oriental Studies Volume18.Princeton University Press,1960.
[7] R.Rashed, The Development of Arabic Mathematics: Between Arithmetic andAlgebra[M]. A F W Armstrong (trans). Kluwer Academic Publishers,1994.
[8] Paul Luckey，Die Rechenkunst bei Gamsid b.Masud al-Kasi[M]，Wiesbaden: F,Steiner,1951.
[9] Jean-Luc Chabert. A history of Alogorithms: from the Pebble to the Microchip[M].Springer,1999.
[10] Sesiano, Jacques,“Abū Kāmil”, Encyclopedia of the history of science[M],technology and medicine in non-weatern cultures. Springer:1997.
[11] Sesiano, Jacques, An introduction to the history of algebra: solving equationsfrom Mesopotamian times to Renaissance[M]. AMS Bookstore,2009.
[12] Helaine Selin(ed.), Mathematics Across Cultures: The History of Non-WesternMathematics[M]. Kluwer Academic Publishers. ISBN0-7923-6481-3.
[13]郭书春,九章算术译注[M].上海:上海古籍出版社,2007.
[14](美)卡兹著,李文林,邹建成,胥鸣伟译.数学史通论(第二版)[M].北京:高等教育出版社,2003.
[15]纪志刚,《孙子算经》、《张邱建算经》、《夏侯阳算经》导读[M].武汉:湖北教育出版社,1999.
[16]李继闵,东方数学典籍《九章算术》及刘徽注研究[M].西安:陕西人民教育出版社,1990.
[17]李继闵,《九章算术》导读与译注[M].西安:山西科学技术出版社,1998.
[18]李文林,数学史概论(第二版)[M].北京:高等教育出版社,2002.
[19](美)莫里斯克莱因著,张理京,张锦炎,江泽涵译,古今数学思想[M].上海:上海科学技术出版社,2002.
[20](美)霍华德伊夫斯（著），欧阳绛（译），数学史概论（第六版）[M].哈尔滨：哈尔滨工业大学出版社，2009.
[21]李迪主编.中外数学史教程[M].福州:福建教育出版社.1993.
[22]（美）卡尔B鲍耶（CARL.B.BOYER著）,（美）尤塔C梅兹巴赫（UTA.C.MERABACH修订）,秦传安（译）.数学史（上册）[M].北京:中央编译出版社.2012.
[1] E.S.Kenndy, A Letter of Jashī aP-KāW ī XS H W FaX V[J]. Scientific Research andPersonalities at a Fifteen Century Court. Orientalia29:P191-213,1960.
[2] M Bagheri, A newly found letter of al-KāW ī on scientific life in Samarkand[J],Historia Math.24(3)(1997),241-256.
[3] Trifkovic, Serge, T GSP R A S IWPaQ W a M]X [J], extract from the bookThe Sword of the Prophet: A Politically-Incorrect Guide to Islam; reproduced inwww.frontpagemag.com, November15,2002.
[4] P.Luckey, Die Ausziehung des n-ten Wurzel und der binomische Lehrsatz in derislamischen Mathemarik[J], Math, Annalen120,1948, P217-274.
[5] Abdul-Kader Dakhel, Edited by Wasf ī A.Hijab and E.S.Kemmedy, TheExtraction of the n-th root in the sexagesimal notation[D], a study of chapter5,treatise III of by Jamshīd Ghiyāth al-Dīn al-Kāshī with translation and commentatry,Beirut: American University,1960.
[6] R Rashed, Combinatorial analysis, numerical analysis, Diophantine analysis andnumber theory[A]. R Rashed: Encyclopedia of the History of Arabic Science[C]. NewYork: Routledge,1996.
[7] Jeffrey A.Oaks, Haitham M.Alkhateeb, Māl, enunciations, and the prehistory ofArabic algebra [J]. History Mathematica32(2005).
[8] Liu Dun. A homecoming stranger: transmission of the method of the Double-False-Position and the story of the Hiero s Crown. From China to Paris:2000TearsTransmission of Mathematical Ideas[C]. Germany: Franz Steiner Verlag WiesbadenGmbH,2002. P157-166.
[9]钱宝琮,《九章算术》盈不足术流传欧洲考.钱宝琮科学史论文集[C].北京:科学出版社，1983. P83-96.
[10]钱宝琮,增乘开方法的历史发展[A].见:钱宝琮.宋元数学史论文集[C].北京:科学出版社,1966. P36-59.
[11]杜石然,试论宋元时期中国和伊斯兰国家间的数学交流[A].见:钱宝琮.宋元数学史论文集[C].北京:科学出版社,1966.
[12]杜石然,再论中国与阿拉伯国家间的数学交流[J].自然科学史研究,1984,3(4),P294-303.
[13]李继闵.刘徽关于无理数的论述[J].西北大学学报（自然科学版）.1989,19(1),P1-4.
[14]赵继伟,《大术》研究[博士论文].西安:西北大学.2005.
[15]包芳勋,阿拉伯代数方程求解几何方法的比较研究[J].自然科学史研究,1997,16(2), P119-129.
[16]包芳勋,孙庆华,阿拉伯与中国代数方程数值解法的比较研究──阿尔·徒思与宋元数学家比较的个案分析[J].自然科学史研究,1999,18(3), P196-205.
[17]赵栓林.对《代数学》和《代数术》术语翻译的研究[硕士论文].呼和浩特:内蒙古师范大学.2006.
[18]郑方磊,许凯《算术三编》研究[硕士论文].上海:上海交通大学.2007.
[19]朱一文,百鸡术的历史研究[硕士论文].上海:上海交通大学.2008.
[20]马丁玲,斐波那契《计算之书》研究[博士论文].上海:上海交通大学.2009.
[21]郭园园,花拉子米《代数学》的比较研究[硕士论文].天津:天津师范大学.2009.
[22]郭园园,《算术之钥》与中算若干问题的比较研究[J].自然科学史研究,2012,31(1),P107-128.
[23]郭园园,《计算之书》中“elchataym”算法探源[J].内蒙古师范大学学报（自然科学汉文版）,2013,42(1),P1-6.
[24]杨淑辉,张妍,数学史家尤什克维奇的杰出贡献[J].辽宁师范大学学报(自然科学版),2010,2(4),P161-164.
2（英）F.Robinson.Cambridge Illustrated History of Islmic World[M].北京:世界知识出版社.2005. P299-302.
3（美）霍华德伊夫斯（著），欧阳绛（译），数学史概论（第六版）[M].哈尔滨：哈尔滨工业大学出版社，2009. P237-238.
5Joseph, George Gheverghese, The Crest of the Peacock: Non-European roots of Mathematics, Harmondsworth:Penguin,1992.P344.
6（美）Katz, Victor J., A History of Mathematics: an Introduction, Reading MA: Addison-Wesley,1998.P240.
7吴文俊,中国数学史的新研究（王志健译）.自然杂志，1989,12（7），P.546-550.
8吴文俊,关于研究数学在中国的历史与现状--《东方数学典籍<九章算术>及其刘徽注研究》序言,自然辩证法通讯,1990,68（1），P37-39.
9李文林，丝绸之路数学名著译丛（导言），北京：科学出版社，2008.Pxi-xii.
10（英）F.Rosen (trs.), Muhammad ibn Musa Al-Khwarizmi: Algebra.London,1831.
11Jamshid al-Kāshī. Miftah al-Hisab (Key to Arithmetic)[M]. A S al-Demerdash.M H al-Cheikh (eds). Cairo: D al--,1967.
12Al-Khwarzmi’s Algebra[M], A Reprint of the Rosen s Translation of1831,Pakistan HijraCouncil,Islamlabad,1989,No.80of the Great Books Project.
13杨淑辉,张妍.数学史家尤什克维奇的杰出贡献[J].辽宁师范大学学报(自然科学版),2010,2(4):161-164.
15Al-，B.A.Rosenfeld（译）,Rosenfeld&A.P.Youschkevitch（评注,Mathematicheskie traktaty，Istorikomatematicheskie issledovaniya Volumu VII，1954.其中《算术之钥》的译文位于PP.13-326.
16Al-，B.A.Rosenfeld（译）,V.S.Segal&A.P.Youschkevitch（编辑）,Rosenfeld和A.P.Youschkevitch（评注），Klyuch Arifemetiki; Traktat ob Okruzhnosti (The Key to Arithmetic; The Treatise on the Circumference)，Moscow：Gosudarstvennoe Izdatel stvo Tekhniko-teoreticheskoi Literatury，1956.其中《算术之钥》的译文位于PP.7-262，《论圆周》的译文位于PP.263-319，随后是评注，最后于PP.428-568附有《算术之钥》莱顿手稿的照片。
17R Rashed.The Development of Arabic Mathematics: Between Arithmetic and Algebra[M]. A F W Armstrong(trans). Kluwer Academic Publishers,1994.p.201, note8.
19Jamshid al-Kāshī, al-Nabulsi,Miftah al-Hisab (Key to Arithmetic), Damascus, Syris: Ministry of High Education,1977.
20（德）Paul Luckey，Die Rechenkunst bei Gamsid b.Masud al-Kasi，Wiesbaden: F, Steiner,1951.
21Roshdi.Rashed, Entre arithmetique at algebra: recherché sur I’histoire des mathematiques arabes, Paris: lesBelles letters,1984.
22（法）Roshdi Rashed, trans, by A.F.W.Armstrong, The development of Arabic mathematics: between arithmeticand algebra, Dordrecht: Boston; London: Kluwer academic,1994.
23P.Luckey, Die Ausziehung des n-ten Wurzel und der binomische Lehrsatz in der islamischen Mathemarik, Math,Annalen120,1948, pp.217-274.
24（美）Abdul-Kader Dakhel, Edited by Wasfī A.Hijab and E.S.Kemmedy, The Extraction of the n-th root in thesexagesimal notation, a study of chapter5, treatise III of by Jamshīd Ghiyāth al-Dīn al-Kāshī with translationand commentatry, Beirut: American University,1960.
25R Rashed.The Development of Arabic Mathematics: Between Arithmetic and Algebra[M]. A F W Armstrong
(trans). Kluwer Academic Publishers,1994.P143,Note,10.
26萨马瓦尔的主要著作有两本：一本是《光辉代数》1972年由S.Ahmd和R.Rashed共同校订出版，其为阿拉伯文版并附有法文简介（S.Ahmd and R.Rashed eds., al-Samaw al, al-Bahir en algebra, Damascus,1972）；另外一本是名为al-Qiwami fi al-Hisab al-Hindi(1172-1173)的算术著作，此书全文已经遗失，但是其第三卷的手稿得以保存，详见R Rashed.The Development of Arabic Mathematics: Between Arithmetic and Algebra[M]. AFW Armstrong (trans). Kluwer Academic Publishers,1994.p143, Note8.
27P Luckey, Die Rechnenkunst bei Gamsid b. Masud al-Kasi (Wiesbaden,1951).
28R Rashed, The development of Arabic mathematics: between arithmetic and algebra (London,1994).P127-134.
30B A Rosenfeld, A P Youschkevitch, Biography in Dictionary of Scientific Biography (New York1970-1990).P256.
31E.S.Kennedy, The Planetary Equatorium of Jamshid Ghiyath al-Din al-Kāshī[M].Princeton Oriental StudiesVolume18.Princeton University Press,1960,P1.
33B A Rosenfeld, A P Youschkevitch, Biography in Dictionary of Scientific Biography (New York1970-1990). P256.
34M Bagheri, A newly found letter of al-on scientific life in Samarkand, Historia Math.24(3)(1997),241-256.
35P.Luckey,Der Lehrbrief Uber den Kreisumfang von Gamsid b. Masud al-Kasi, Klasse fur Mathematik undallgemeine Naturwissenschaften Jahrgang,1950,Nr.6.
36B.A.Rosenfeld, A P Youschkevitch, Biography in Dictionary of Scientific Biography (New York1970-1990). P257.
37赵栓林.对《代数学》和《代数术》术语翻译的研究[硕士论文].2006:内蒙古师范大学.
38艾约瑟.阿尔热巴喇附考,中西闻见录第七号.1873.
39（美）卡尔B鲍耶（CARL.B.BOYER著）,（美）尤塔C梅兹巴赫（UTA.C.MERABACH修订）,秦传安（译）.数学史（上册）[M].北京:中央编译出版社.2012.P256.
50R.Rashed et. B.Vahabzaded. Al-Khayyam Mathematicien[M]. Paris: Librairie Scientifique et Technique AlbertBlanchard,1999. P125-129.
51S.Ahmd and R.Rashed eds., al-Samaw al, al-Bahir en algebra, Damascus,1972.P119.
52（意）斐波那契原著，（美）西格尔英译，纪志刚、汪晓勤、马丁玲、郑方磊译.计算之书.北京:科学出版社，2007.P513.
53Jamshid al-Kāshī. Miftah al-Hisab (Key to Arithmetic)[M]. A S al-Demerdash.M H al-Cheikh (eds). Cairo: D al--,1967.P211.
54S.Ahmd and R.Rashed eds., al-Samaw al, al-Bahir en algebra[M], Damascus,1972.P140.
61Jamshid al-Kāshī. Miftah al-Hisab (Key to Arithmetic)[M]. A S al-Demerdash.M H al-Cheikh (eds). Cairo: D al--,1967. P225.
69Jamshid al-Kāshī. Miftah al-Hisab (Key to Arithmetic)[M]. A S al-Demerdash.M H al-Cheikh (eds). Cairo: D al--,1967.P234.
82T.L.Heath, The Thirteen Books of Euclid s Elements[M], New York: Dover Publications, INC.1956. Volume I, P71.
83Jamshid al-Kāshī. Miftah al-Hisab (Key to Arithmetic)[M]. A S al-Demerdash.M H al-Cheikh (eds). Cairo: D al--,1967.P264.
87Jamshid al-Kāshī. Miftah al-Hisab (Key to Arithmetic)[M]. A S al-Demerdash.M H al-Cheikh (eds). Cairo: D al-al-,1967.P268.
137S.Ahmd and R.Rashed eds., al-Samaw al, al-Bahir en algebra, Damascus,1972.P63.
159Jamshid al-Kāshī. Miftah al-Hisab (Key to Arithmetic)[M]. A S al-Demerdash.M H al-Cheikh (eds). Cairo: D1a6l-0-,1967.P66-67.
163R Rashed.The Development of Arabic Mathematics: Between Arithmetic and Algebra[M]. A F W Armstrong
(trans). Kluwer Academic Publishers,1994.P109-110.
165R Rashed. Combinatorial analysis, numerical analysis, Diophantine analysis and number theory[A]. R Rashed:Encyclopedia of the History of Arabic Science[C]. New York: Routledge,1996.P383.
170Jamshid al-Kāshī. Miftah al-Hisab (Key to Arithmetic)[M]. A S al-Demerdash.M H al-Cheikh (eds). Cairo: D al--,1967.P73.
177R Rashed.The Development of Arabic Mathematics: Between Arithmetic and Algebra[M]. A F W Armstrong(trans). Kluwer Academic Publishers,1994.P136.
178Al-Samaw al. al-Bahir en algebra[M]. S Ahmd.R Rashed (eds). Damascus:Imp. De I Universite deDamas,1972.P111.
179Jamshid al-Kāshī. Miftah al-Hisab (Key to Arithmetic)[M]. A S al-Demerdash.M H al-Cheikh (eds). Cairo: D al--,1967. P73.
182P Luckey. Die Ausziehung des n-ten Wurzel und der binomische Lehrsatz in der islamischen Mathemarik[A].Springer:Math Annalen120,1948.217-274.
183杜石然.试论宋元时期中国和伊斯兰国家间的数学交流[A].见:钱宝琮.宋元数学史论文集[C].北京:科学出版社,1966.P258.
184杜石然.再论中国与阿拉伯国家间的数学交流[J].自然科学史研究,1984,3(4):294-303.
185钱宝琮.增乘开方发的历史发展[A].见:钱宝琮.宋元数学史论文集[C].北京:科学出版社,1966.36-44.
186郭书春.九章算术译注[M].上海:上海古籍出版社,2007.133.
188李继闵.刘徽关于无理数的论述[J].西北大学学报（自然科学版）.1989,19(1),P1-4.
189钱宝琮.增乘开方发的历史发展[A].见:钱宝琮.宋元数学史论文集[C].北京:科学出版社,1966.P37-38.
190杜石然.试论宋元时期中国和伊斯兰国家间的数学交流[A].见:钱宝琮.宋元数学史论文集[C].北京:科学出版社,1966.P259.
194S.Ahmd and R.Rashed eds., al-Samaw al, al-Bahir en algebra, Damascus,1972.P160.
208（意）斐波那契原著，（美）西格尔英译，纪志刚、汪晓勤、马丁玲、郑方磊译.计算之书[M].北京：科学出版社，2007.P526.
214郑方磊.许凯《算术三编》研究.上海交通大学硕士论文.2007.p47.
215马丁玲.斐波那契《计算之书》研究.上海交通大学博士论文.2009.p128.
216钱宝琮.《九章算术》盈不足术流传欧洲考.《钱宝琮科学史论文集》.北京：科学出版社，1983.P83-96.
217杜石然.试论宋元时期中国和伊斯兰国家间的数学交流《.宋元数学史论文集》.科学出版社,1966.P241-265.
218郭书春，刘钝校点.《算经十书》（一）.沈阳：辽宁教育出版社，1998.
219（明）李之藻（演）,（意）利玛窦（授）.同文算指.郭书春.中国科学技术史典籍通汇.数学卷.第4册.郑
222Sesiano, Jacques, Abū Kāmil, Encyclopedia of the history of science, technology and medicine innon-weatern cultures. Springer1997-07-31. P4-5.
223Sesiano, Jacques, An introduction to the history of algebra: solving equations from Mesopotamian times toRenaissance. AMS Bookstore,2009. P75-76.
224Helaine Selin(ed.), Mathematics Across Cultures: The History of Non-Western Mathematics. Kluwer Academic
235Jamshid al-Kāshī. Miftah al-Hisab (Key to Arithmetic)[M]. A S al-Demerdash.M H al-Cheikh (eds). Cairo: D al--,1967.P243.
242朱一文.百鸡术的历史研究.上海交通大学硕士论文.2008.P29-30.
243张丘建（南北朝）.张丘建算经[A].中国科学技术典籍通汇数学卷[Z].第1册.郑州：河南教育出版社，1993.
244甄鸾（南北朝）.数术记遗[A].郭书春，刘钝.算经十书[Z].沈阳：辽宁教育出版社，1998.
245（意）斐波那契原著，（美）西格尔英译，纪志刚、汪晓勤、马丁玲、郑方磊译.计算之书.北京:科学出版社，2007.P261-262.
248（意）斐波那契原著，（美）西格尔英译，纪志刚、汪晓勤、马丁玲、郑方磊译.计算之书.北京:科学出版社，2007.P265-266.
259Al-Khwārizmī,Edited with translation and commentary by Roshdi Rashed, The Beginnings of Algebra[M], SAQI,Landon,2009.
260Berggren, J. Lennart,"Mathematics in Medieval Islam", in Victor J. Katz, The Mathematics of Egypt,Mesopotamia, China, India, and Islam[M]: A Sourcebook (Second ed.)[M], Princeton, New Jersey: PrincetonUniversity,2007.
261J.L. Berggren，Episodes in the mathematics of medieval Islam. Springer,2003.
262Jamshid al-Kāshī, Miftah al-Hisab (Key to Arithmetic)[M]. A S al-Demerdash.M H al-Cheikh (eds).Cairo: Dār al-kātib al-arabī,1967.
263郭园园，《算术之钥》与中算若干问题的比较研究[J].自然科学史研究,2012,31(1),P107-128.
[1] Al-Khwārizmī,Edited with translation and commentary by Roshdi Rashed, TheBeginnings of Algebra[M], SAQI, Landon,2009.
[2] F.Woepcke, E\XVa X Y FaO V, XVa X 'aP bV TaV AbSY B OV MS aQQ B RAlhacan Alkarkhi[M], Reprint of the1853original (Hildesheim,1982).
[3] S.Ahmd and R.Rashed eds., al-Samaw’al, al-Bahir en algebra[M], Damascus,1972.
[4] Jamshid al-Kāshī, Miftah al-Hisab (Key to Arithmetic)[M]. A S al-Demerdash.MH al-Cheikh (eds). Cairo: Dār al-kātib al-arabī,1967.
[5] Jamshid al-Kāshī,al-Nabulsi,Miftah al-Hisab (Key to Arithmetic)[M], Damascus,Syris: Ministry of High Education,1977.
[6] R.Rashed et. B.Vahabzaded, Al-Khayyam Mathematicien[M]. Paris: LibrairieScientifique et Technique Albert Blanchard,1999.
[7] Jacques Sesiano, Books IV to VII of Diophantus Arithmetica in the Arabictranslation attributed to Quatā ibn Lūqā[M], Sources in the History of Mathematicsand physical Sciences3. Springer-Verlag New York, Heidelberg, Berlin.1982.
[8] Kamal al-Din al-Farisi, Asas al-qawa ld fi usul al-fawa ld (The base of the rules inthe principles of uses)[M], edited by M.Mawaldi, Cairo: The Institute if ArabicManuscripts,1994.
[9] L.E.SIGLER, Fibonacci s Liber Abaci (A Translation into Modern English ofLeonardo Pisano’s Book of Calculation)[M]. Springer,2003.
[10]（意）斐波那契原著,（美）西格尔英译,纪志刚,汪晓勤,马丁玲,郑方磊译.计算之书[M].北京:科学出版社,2007.
[11](印)婆什伽罗原著,（日）林隆夫译注,徐泽林,周畅,张建伟译,莉拉沃蒂[M].北京:科学技术出版社,2006.
[12]张丘建（南北朝）,张丘建算经[A].中国科学技术典籍通汇数学卷[Z].第1册.郑州：河南教育出版社，1993.
[13]甄鸾（南北朝）,数术记遗[A].郭书春,刘钝.算经十书[Z].沈阳：辽宁教育出版社，1998.
[14]（明）李之藻,（意）利玛窦[A].同文算指.郭书春.中国科学技术史典籍通汇.数学卷[Z].第4册.郑州：河南教育出版社,1993.
[1] Berggren, J. Lennart,"Mathematics in Medieval Islam", in Victor J. Katz, TheMathematics of Egypt, Mesopotamia, China, India, and Islam[M]: A Sourcebook(Second ed.)[M], Princeton, New Jersey: Princeton University,2007.
[2] P.Luckey, Der Lehrbrief Uber den Kreisumfang von Gamsid b. Masud al-Kasi,Klasse fur Mathematik und allgemeine Naturwissenschaften Jahrgang[M],1950,Nr.6.
[3] Joseph, George Gheverghese, The Crest of the Peacock: Non-European roots ofMathematics[M], Harmondsworth: Penguin,1992.
[4] Katz, Victor J., A History of Mathematics: an Introduction[M], Reading MA:Addison-Wesley,1998.
[5] B A Rosenfeld, A P Youschkevitch, Biography in Dictionary of ScientificBiography [M](New York1970-1990).
[6] E.S.Kennedy, The Planetary Equatorium of Jamshid Ghiyath al-Din al-Kāshī[M].Princeton Oriental Studies Volume18.Princeton University Press,1960.
[7] R.Rashed, The Development of Arabic Mathematics: Between Arithmetic andAlgebra[M]. A F W Armstrong (trans). Kluwer Academic Publishers,1994.
[8] Paul Luckey，Die Rechenkunst bei Gamsid b.Masud al-Kasi[M]，Wiesbaden: F,Steiner,1951.
[9] Jean-Luc Chabert. A history of Alogorithms: from the Pebble to the Microchip[M].Springer,1999.
[10] Sesiano, Jacques,“Abū Kāmil”, Encyclopedia of the history of science[M],technology and medicine in non-weatern cultures. Springer:1997.
[11] Sesiano, Jacques, An introduction to the history of algebra: solving equationsfrom Mesopotamian times to Renaissance[M]. AMS Bookstore,2009.
[12] Helaine Selin(ed.), Mathematics Across Cultures: The History of Non-WesternMathematics[M]. Kluwer Academic Publishers. ISBN0-7923-6481-3.
[13]郭书春,九章算术译注[M].上海:上海古籍出版社,2007.
[14](美)卡兹著,李文林,邹建成,胥鸣伟译.数学史通论(第二版)[M].北京:高等教育出版社,2003.
[15]纪志刚,《孙子算经》、《张邱建算经》、《夏侯阳算经》导读[M].武汉:湖北教育出版社,1999.
[16]李继闵,东方数学典籍《九章算术》及刘徽注研究[M].西安:陕西人民教育出版社,1990.
[17]李继闵,《九章算术》导读与译注[M].西安:山西科学技术出版社,1998.
[18]李文林,数学史概论(第二版)[M].北京:高等教育出版社,2002.
[19](美)莫里斯克莱因著,张理京,张锦炎,江泽涵译,古今数学思想[M].上海:上海科学技术出版社,2002.
[20](美)霍华德伊夫斯（著），欧阳绛（译），数学史概论（第六版）[M].哈尔滨：哈尔滨工业大学出版社，2009.
[21]李迪主编.中外数学史教程[M].福州:福建教育出版社.1993.
[22]（美）卡尔B鲍耶（CARL.B.BOYER著）,（美）尤塔C梅兹巴赫（UTA.C.MERABACH修订）,秦传安（译）.数学史（上册）[M].北京:中央编译出版社.2012.
[1] E.S.Kenndy, A Letter of Jashī aP-KāW ī XS H W FaX V[J]. Scientific Research andPersonalities at a Fifteen Century Court. Orientalia29:P191-213,1960.
[2] M Bagheri, A newly found letter of al-KāW ī on scientific life in Samarkand[J],Historia Math.24(3)(1997),241-256.
[3] Trifkovic, Serge, T GSP R A S IWPaQ W a M]X [J], extract from the bookThe Sword of the Prophet: A Politically-Incorrect Guide to Islam; reproduced inwww.frontpagemag.com, November15,2002.
[4] P.Luckey, Die Ausziehung des n-ten Wurzel und der binomische Lehrsatz in derislamischen Mathemarik[J], Math, Annalen120,1948, P217-274.
[5] Abdul-Kader Dakhel, Edited by Wasf ī A.Hijab and E.S.Kemmedy, TheExtraction of the n-th root in the sexagesimal notation[D], a study of chapter5,treatise III of by Jamshīd Ghiyāth al-Dīn al-Kāshī with translation and commentatry,Beirut: American University,1960.
[6] R Rashed, Combinatorial analysis, numerical analysis, Diophantine analysis andnumber theory[A]. R Rashed: Encyclopedia of the History of Arabic Science[C]. NewYork: Routledge,1996.
[7] Jeffrey A.Oaks, Haitham M.Alkhateeb, Māl, enunciations, and the prehistory ofArabic algebra [J]. History Mathematica32(2005).
[8] Liu Dun. A homecoming stranger: transmission of the method of the Double-False-Position and the story of the Hiero s Crown. From China to Paris:2000TearsTransmission of Mathematical Ideas[C]. Germany: Franz Steiner Verlag WiesbadenGmbH,2002. P157-166.
[9]钱宝琮,《九章算术》盈不足术流传欧洲考.钱宝琮科学史论文集[C].北京:科学出版社，1983. P83-96.
[10]钱宝琮,增乘开方法的历史发展[A].见:钱宝琮.宋元数学史论文集[C].北京:科学出版社,1966. P36-59.
[11]杜石然,试论宋元时期中国和伊斯兰国家间的数学交流[A].见:钱宝琮.宋元数学史论文集[C].北京:科学出版社,1966.
[12]杜石然,再论中国与阿拉伯国家间的数学交流[J].自然科学史研究,1984,3(4),P294-303.
[13]李继闵.刘徽关于无理数的论述[J].西北大学学报（自然科学版）.1989,19(1),P1-4.
[14]赵继伟,《大术》研究[博士论文].西安:西北大学.2005.
[15]包芳勋,阿拉伯代数方程求解几何方法的比较研究[J].自然科学史研究,1997,16(2), P119-129.
[16]包芳勋,孙庆华,阿拉伯与中国代数方程数值解法的比较研究──阿尔·徒思与宋元数学家比较的个案分析[J].自然科学史研究,1999,18(3), P196-205.
[17]赵栓林.对《代数学》和《代数术》术语翻译的研究[硕士论文].呼和浩特:内蒙古师范大学.2006.
[18]郑方磊,许凯《算术三编》研究[硕士论文].上海:上海交通大学.2007.
[19]朱一文,百鸡术的历史研究[硕士论文].上海:上海交通大学.2008.
[20]马丁玲,斐波那契《计算之书》研究[博士论文].上海:上海交通大学.2009.
[21]郭园园,花拉子米《代数学》的比较研究[硕士论文].天津:天津师范大学.2009.
[22]郭园园,《算术之钥》与中算若干问题的比较研究[J].自然科学史研究,2012,31(1),P107-128.
[23]郭园园,《计算之书》中“elchataym”算法探源[J].内蒙古师范大学学报（自然科学汉文版）,2013,42(1),P1-6.
[24]杨淑辉,张妍,数学史家尤什克维奇的杰出贡献[J].辽宁师范大学学报(自然科学版),2010,2(4),P161-164.