關於中國古代籌算捷法和珠算中留頭乘的幾個問題

Several Issues Pertaining to Liutou Cheng in Ancient Abacus Arithmetic

牛騰(Niu Teng) ； 鄒大海(Zou Dahai)

《漢學研究》 39卷1期 (2021/03) Pp. 133-160

摘要

留頭乘是元代以來籌算捷法和珠算中乘法計算的常用方法。前人對留頭乘的口訣和歷史曾有同異互見的闡釋，留下一些疑難問題。本文對古代留頭乘的方法和表達方式進行了重新梳理，通過仔細分析古算書中關於留頭乘的說明、口訣和例題，證明李培業等關於《算學啟蒙》中留頭乘口訣的解釋是正確的，而潘逢禧等人的解釋不符合口訣的本意。本文還證明《九章算法比類大全》等算書中「乘法」的說明與口訣並不矛盾，只是元明算書的例題有的與之吻合，有的與之不吻合；這些反映了古代算書編輯的複雜性。本文指出，《九章算法比類大全》中關於留頭乘的內容具有更早的淵源，《數學通軌》中有關留頭乘的內容並不直接源於《詳明算法》。

關鍵字

留頭乘 ； 新解釋 ； 《算學啟蒙》 ； 《九章算法比類大全》 ； 《數學通軌》 ； 《詳明算法》

並列摘要

Liutou cheng留頭乘was a usual method of multiplication in rapid calculation by counting rods and in abacus arithmetic since the Yuan dynasty. Scholars have held different interpretations of the mnemonic rhyme and the history of liutou cheng, leaving several difficult questions awaiting our attention. The present article reexamines the development of liutou cheng and its methods of expression. By carefully analyzing the provided mnemonic rhymes, examples, and related captions concerning liutou cheng in ancient mathematics books, the article proves that the interpretations asserted by Li Peiye李培業and other scholars on the mnemonic rhyme of liutou cheng in Suanxue qimeng算學啟蒙(Introduction to Mathematics, 1299) are correct, whereas the explanations of Pan Fengxi潘逢禧and others do not accord with the original meaning of the mnemonic rhyme. This paper also proves that contrary to what several scholars have presumed, the caption under the title ＂chengfa＂ 乘法(multiplication) in Jiuzhang suanfa bilei daquan九章算法比類大全(Comprehensive Collection of Analogical Methods of the Nine Chapters on Mathematical Methods, 1450) and several other books is not contradictory with the mnemonic rhyme; rather, among the examples in the mathematics books of the Ming dynasty, some are merely coincident with the mnemonic rhyme and caption, whereas others are not. These facets reflect the complexity of the compilations of ancient mathematics books. Finally, the present paper indicates that the content of liutou cheng found within Jiuzhang suanfa bilei daquan is based on earlier sources, and that the part regarding liutou cheng in Shuxue tonggui數學通軌(Clear Path to Mathematics, 1578) does not originate directly from Xiangming suanfa詳明算法(Detailed and Clear Methods of Mathematics, earlier than 1369).

並列關鍵字

liutou cheng ； new interpretation ； Suanxue qimeng (Introduction to Mathematics) ； Jiuzhang suanfa bilei daquan (Comprehensive Collection of Analogical Methods of the Nine Chapters on Mathematical Methods) ； Shuxue tonggui (A Clear Path to Mathematics) ； Xiangming suanfa (Detailed and Clear Methods of Mathematics)

參考文獻

宋．謝察微，《謝察微算經》，收入郭書春主編，《中國科學技術典籍通匯．數學卷．第1 分冊》，鄭州：河南教育出版社，1993。

宋．楊輝，《楊輝算法》，收入郭書春主編，《中國科學技術典籍通匯．數學卷．第1分冊》，鄭州：河南教育出版社，1993。

元．朱世傑，《算學啓蒙》，收入郭書春主編，《中國科學技術典籍通匯．數學卷．第1 分冊》，鄭州：河南教育出版社，1993。

元．賈亨，《算法全能集》，收入郭書春主編，《中國科學技術典籍通匯．數學卷．第1 分冊》，鄭州：河南教育出版社，1993。

元．安止齋、何平子，《詳明算法》，收入郭書春主編，《中國科學技術典籍通匯．數學卷．第1 分冊》，鄭州：河南教育出版社，1993。