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自然科学史研究  2019, Vol. 38 Issue (2): 172-188    
论文     
印度库塔卡详解及其与大衍总数术比较新探
吕鹏, 纪志刚
上海交通大学科学史与科学文化研究院,上海 200240
An In-depth Study on Indian kut·t·aka and Comparison with the Chinese Da-yan Rule
LÜ Peng, JI Zhigang
School of History and Culture of Science, Shanghai Jiao Tong University, Shanghai 200240, China
 全文: PDF(1462 KB)  
摘要： 古代印度数学中的"库塔卡"既指一次不定问题,又指解决此类问题时的一套算法。自从它出现于5世纪《阿耶波多历算书》之后,一直是印度数学研究的主要论题之一。通过研读梵语原典,在说明库塔卡的产生、发展、特点和有效性之后,将它与中算大衍总数术做比较,确认后者的关键部分大衍求一术与库塔卡的互除原理、迭代计算和数字阵型上具有一定的相似性,大衍求一术问题其实是特殊类型的库塔卡,库塔卡的解题能力实际等同于大衍总数术。然而,库塔卡与大衍总数术在算法结构和历史发展上都有较大差异,并且由于库塔卡有一套约化和联立规则,加上印度数学家能熟练使用0和负数进行计算,因此在处理同余问题上库塔卡较大衍总数术要更为简单快捷。
关键词： 库塔卡;  阿耶波多;  印度数学史;  大衍求一术;  大衍总数术    
Abstract: The word Kut·t·aka means the problem of first order indefinite analysis and also the operational algorithm of this kind of problem in the works of ancient Indian mathematics. After it first appeared inĀryabhat·a's Āryabhat·īya (5th century A.D.), Kut·t·aka was an important topic for Indian mathematicians. Based on Sanskrit texts, we discuss aspects of the origin, improvement, main features and effectiveness of the Kut·t·aka algorithm. Then, comparing Kut·t·aka with the Chinese Dayan-Zongshu method, we confirm the similarity between Kut·t·aka and the Da-yan Rule on the computation of the Euclidean Algorithm, as well as in their systematic design (i.e., iterative computation) and graphically (i.e., the creeper of remainders and the square of manipulating numbers). In fact, the Dayan-Qiuyi method is a special kind of Kut·t·aka; the power of the Kut·t·aka is equivalent to the Dayan Rule. However, the two are quite different in the whole structure of the algorithm and in historical development. Moreover, the Kut·t·aka method seems to be more general, simpler and easier because of a series of rules of reduction and continuity.
Key words: Kut·t·aka    Āryabhat·a    history of Indian mathematics    Dayan-Qiuyi method    Da-yan Rule (Dayan-Zongshu-shu)
收稿日期: 2017-08-07 出版日期: 2022-06-18
ZTFLH:  O11  
基金资助: 教育部人文社会科学基金青年项目"基于梵语原典的印度中世历算书《婆罗摩修正体系》研究"(18YJCZH121)。
作者简介: 吕鹏,1983年生,上海人,助理教授,主要研究方向为印度数理天文学史。纪志刚,1956年生,江苏邳州人,教授,主要研究方向为数学史和中外数学文化交流史。
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引用本文:

吕鹏, 纪志刚. 印度库塔卡详解及其与大衍总数术比较新探[J]. 自然科学史研究, 2019, 38(2): 172-188.

LÜ Peng, JI Zhigang. An In-depth Study on Indian kut·t·aka and Comparison with the Chinese Da-yan Rule. Studies in the History of Natural Sciences, 2019, 38(2): 172-188.

链接本文:

http://www.shns.ac.cn/CN/    或    http://www.shns.ac.cn/CN/Y2019/V38/I2/172

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