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Ancient Chinese Mathematics: the (Jiu Zhang Suan Shu) Vs Euclid's Elements

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Ancient Chinese Mathematics: the (Jiu Zhang Suan Shu) Vs Euclid's Elements PERGAMON International Journal of Engineering Science 36 (1998) 1339±1359 Ancient Chinese mathematics: the (Jiu Zhang Suan Shu) vs Euclid's Elements. Aspects of proof and the linguistic limits of knowledge Joseph W. Dauben a, b, * aDepartment of History, Herbert H. Lehman College, City University of New York, 250 Bedford Park Blvd. West, Bronx, NY 10468, USA bPh.D. Program in History, The Graduate Center, City University of New York, 33 West 42nd Street, New York, NY 10036, USA Abstract The following is a preliminary and relatively brief, exploratory discussion of the nature of early Chinese mathematics, particularly geometry, considered largely in terms of one speci®c example: the (Gou-Gu) Theorem. In addition to drawing some fundamental comparisons with Western traditions, particularly with Greek mathematics, some general observations are also made concerning the character and development of early Chinese mathematical thought. Above all, why did Chinese mathematics develop as it did, as far as it did, but never in the abstract, axiomatic way that it did in Greece? Many scholars have suggested that answers to these kinds of questions are to be found in social and cultural factors in China. Some favor the sociological approach, emphasizing for example that Chinese mathematicians were by nature primarily concerned with practical problems and their solutions, and, therefore, had no interest in developing a highly theoretical mathematics. Others have stressed philosophical factors, taking another widely-held view that Confucianism placed no value on theoretical knowledge, which, in turn, worked against the development of abstract mathematics of the Greek sort. While both of these views contain elements of truth, and certainly play a role in understanding why the Chinese did not develop a more abstract, deductive sort of mathematics along Greek lines, a dierent approach is oered here. To the extent that knowledge is transmitted and recorded in language, oral and written, logical and linguistic factors cannot help but have played a part in accounting for how the Chinese were able to conceptualizeÐand think aboutÐmathematics. # 1998 Elsevier Science Ltd. All rights reserved. * Tel.: 011 212 642 2110; Fax: 001 212 642 1963; E-mail: [email protected]. 0020-7225/98/$19.00 # 1998 Elsevier Science Ltd. All rights reserved. PII: S0020-7225(98)00036-6 1340 J.W. Dauben / International Journal of Engineering Science 36 (1998) 1339±1359 1. Jiu Zhang Suan Shu One of the oldest and most in¯uential works in the history of Chinese mathematics is the (Jiu Jang Suan Shu, Nine Chapters on the Art of Mathematics), comprised of nine chapters and hence its title (Fig. 1).1 Traditionally, this work is believed to include some of the oldest mathematical results of Chinese antiquity. Indeed, the origins of the Jiu Jang Suan Shu Fig. 1. Title page, from the Southern Song edition of the (Jiu Zhang Suan Shu, Nine Chapters on the Art of Mathematics), from the copy in the Shanghai Library, facsimile edition: Shanghai: Wen Wu Chu Ban She (Wen Wu Publishing House), 1981. 1 For Chinese editions of the Nine Chapters, see Refs. [2, 3 and 29], along with the detailed studies in Refs. [1] and [4]; there is also a translation into German in Ref. [5]. A French translation of the Nine Chapters, by Karine Chemla and Guo Shu-Chun, is now in preparation and due to be published shortly. J.W. Dauben / International Journal of Engineering Science 36 (1998) 1339±1359 1341 have been ascribed by some to the earliest period of China's recorded history, where fact shades into myth. One tradition says that the Yellow Emperor, Huang Di , who lived in the 27th century BC, charged his minister Li Shou with compiling the Jiu Jang Suan Shu. Unfortunately, the original version of the Nine Chapters no longer exists. One of the ®rst great tragedies in Chinese intellectual history occurred in 213 BC, when the Emperor Qin Shi Huang (221±207 BC, famed for his terra-cotta army at Xi'an Yang ) ordered that all books in the Empire be burned. Although some of the classics may have been surreptitiously preserved, or memorized and later transcribed, the reconstituted texts produced for these ``lost'' early documents likely contained inaccuracies or interpolations introduced by their rescuers. In the case of mathematical knowledge, later innovations and new techniques might well have been incorporated as if they had been part of the original. The subsequent history of the Nine Chapters is nearly as uncertain as its origins. The earliest text we have of the Jiu Zhang Suan Shu was compiled by Zhang Cang sometime in the 2nd century BC, and revised about 100 years later by Geng Shou Chang . Both of these scholars lived in the Western Han Dynasty (206 BC±24 AD), and both were imperial ministers who undertook their reconstructions of the Nine Chapters at a time when there were great eorts being made to restore lost classics of any sort [6]. When Liu Hui ,a mathematician of Wei during the Three Kingdoms Period (220±280 AD), again edited the Jiu Zhang Suan Shu in 263 AD, this time with an extensive commentary, he began a tradition that was repeated after him by Li Chun Feng [ ] in the Tang Dynasty (618±907 AD), who also collated and commented on the book.2 The oldest edition of the Nine Chapters to survive is the wood block printing of the Southern Song Dynasty. Only the ®rst ®ve books (or chapters) are preserved in this edition, which was produced about 1213 AD and is best known from the copy in the Shanghai library. Most other editions are based on the Complete Library of the Four Branches of Literature edited by Dai Zhen of the Qing Dynasty, who copied it from the Great Encyclopedia of the Yong-le Reign Period of the Ming Dynasty (known as the Dai edition). The most famous commentaries are those by Liu Hui (263 AD), Li Chung-Feng (656 AD), and one by Zu Chong-Zhi [ ] (429±500 AD) written during the North and South Dynasties, but now lost.3 2 One of the Great Masters of ancient Chinese mathematics, Liu Hui is all but an enigma in the history of Chinese science. Based on nothing more than philological evidence, it is only possible to say that he may have come from Zouping in the Shandong Province. Based upon a contemporary report, it can at least be said that he com- posed his commentary on the Nine Chapters in 263 AD. All that is known about Liu Hui, in fact, is his well-known commentary on the famous Chinese mathematical classic, the Nine Chapters, and a treatise that Liu Hui wrote him- self, the (Hai Dao Suan Jing, The Sea Island Mathematical Manual). This ``manual'', which takes its name from the ®rst problem devoted to calculating the height and distance of an island at sea, contains only nine problems. It was originally intended as an additional ``tenth'' chapter to the Nine Chapters, but it later came to be regarded as a classic, independently, on its own. For further discussion of The Sea Island Mathematical Manual, see Ref. [7], and the recent English translation in Ref. [8]. For studies speci®cally devoted to Liu Hui, see Refs. [1, 4, 9± 19]. 3 The date of Liu Hui's edition and commentary on the Jiu Zhang Suan Shu is based on a sentence in the manu- script which mentions that it was written in the fourth year of the Jing Yuan reign of King Chen Liu of Wei, which dates it exactly to 263 AD. See Ref. [7]. 1342 J.W. Dauben / International Journal of Engineering Science 36 (1998) 1339±1359 The Jiu Zhang Suan Shu dominated the practice of Chinese administrative clerks for more than a millennium, and yet in its social origins it was closely bound up with the bureaucratic government system, and was consequently devoted to the problems which ruling ocials had to solve. It was also of overwhelming in¯uence on writers in the centuries that followed, and it is no exaggeration to say that virtually all subsequent Chinese mathematics bears its imprint as to both ideas and terminology [20]. It is in this sense that the Nine Chapters may be regarded as a Chinese counterpart to Euclid's Elements, which dominated Western mathematics in the same way the Nine Chapters came to be regarded as the seminal work of ancient Chinese mathematics for nearly two millennia.4 In several important respects, however, the two works are more striking for the dierences they exhibit rather than their similarities. Euclid's text is renowned for the austerity of its axiomatic method, beginning with abstract, idealized de®nitions and proceeding from axioms and postulates to a progressively arranged series of proofs, leading, through the thirteen books that survive, to some remarkable results on the ®ve regular (sometimes called Platonic) polyhedra. The Nine Chapters, on the other hand, is a much more down-to-earth (literally) handbook for the solution of practical problems. However these often led to solutions that were as computationally dicult as they were theoretically subtle. Chinese mathematicians were just as creative in devising new methods as their contemporaries anywhere in the world, particularly in obtaining solutions of simultaneous equations, in which they had no rivals in antiquity.5 2. Mathematics and the Nine Chapters Like earlier works of ancient Chinese mathematics upon which the Jiu Zhang Suan Shu was doubtless based, it presents a series of problems (246 in number) in a question±answer format.6 Those in the ®rst eight chapters deal with such practical concerns as surveying, commercial problems, partnerships and taxation rates.
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