Ji Gu Suan Jing
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The Grand Astrologer’s platform and ramp: Four problems in solid geometry from Wang Xiaotong’s ‘Continuation of ancient mathematics’ (7th century AD)
Tina Su-lyn Lim is a project manager with NNIT (Novo Nordisk Information Technology) in Copenhagen, Denmark. This article formed part of her M.Sc. thesis in Mathematics at the University of Copenhagen, 2006. Donald B. Wagner has taught at the University of Copenhagen, the University of Victoria (British Columbia), and the Technical University of Berlin. He has written widely on the history of science and technology in China; his most recent book is the volume on Ferrous Metallurgy in Joseph Needham’s Science and Civilisation in China. We are grateful to Karine Chemla for drawing our attention to Wang Xiaotong’s interesting text, to Jesper Lützen and Ivan Tafteberg for detailed comments on an earlier version, and to Jia-ming Ying and three anonymous reviewers for comments on earlier drafts of this version. Abstract Wang Xiaotong’s Jigu suanjing is primarily concerned with problems in solid and plane geometry leading to cubic equations which are to be solved numerically by the Chinese variant of Horner’s Method. The problems in solid geometry give the volume of a solid and certain constraints on its dimensions, and the dimensions are required; we translate and analyze four of these. Three are solved using dissections, while one is solved using reasoning about calculations with very little recourse to geometrical considerations. The problems in Wang Xiaotong’s text cannot be seen as practical problems in themselves, but they introduce mathematical methods which would have been useful to administrators in organizing labour forces for public works. Keywords Wang Xiaotong, Jigu suanjing, China, Tang Dynasty, solid geometry Introduction Wang Xiaotong (late 6th–7th century AD) served the Sui and Tang dynasties in posts concerned with calendrical calculations, and presented his book, Jigu suanjing ‘Continuation of ancient mathematics’, to the Imperial court at some time after AD 626. In 656 it was made one of ten official ‘canons’ (jing ) for mathematical education. The book contains 20 problems: one astronomical problem, then 13 on solid geometry, then six on right triangles. All but the first provide extensions of the methods in the mathematical classic Jiuzhang suanshu (perhaps 1st century AD): [2] the solid-geometry methods in Chapter 5 and the right-triangle methods in Chapter 9, [3] thus ‘continuing’ ancient mathematics. All but the first require the extraction of a root of a cubic or (in two cases) quadratic equation. The present article is concerned with problem 2, which in fact consists of four connected but distinct geometric problems. Printed in smaller characters in the text are comments which generally give explanations of the algorithms of the main text. The earliest extant edition states that the comments are by Wang Xiaotong himself, but we have noticed some differences in terminology between the comments and the main text, [4] and feel therefore that the question of the authorship of the comments should be left open. Perhaps most but not all of the comments are by him.
Chinese Mathematics
Mathematics in China emerged independently by the 11th century BC. The Chinese independently developed very large and negative numbers, decimals, a decimal system, a binary system, algebra, geometry, trigonometry. Most scholars believe that Chinese mathematics and the mathematics of the ancient Mediterranean world had developed more or less independently up to the time when the The Nine Chapters on the Mathematical Art reached its final form, while the Writings on Reckoning and Huainanzi preceded it. It is often suggested that some Chinese mathematical discoveries predate their Western counterparts. One example is the Pythagorean theorem. There is some controversy regarding this issue and the precise nature of this knowledge in early China. The Chinese were one of the most advanced in dealing with mathematical computations, and created enormous numbers. Elements of "Pythagorean" science have been found, for example, in one of the oldest Classical Chinese texts (see King Wen sequence). This book was known for all of the mathematical information it contained. Knowledge of Pascal's triangle has also been shown to have existed in China centuries before Pascal, such as by Shen Kuo. Knowledge of Chinese mathematics before 100 BC is somewhat fragmentary, and even after this date the manuscript traditions are obscure. The dating of the use of certain mathematical methods in Chinese history is problematic and disputed. In early times the focus was on astronomy and perfecting the calendar and not on establishing the proof. Many works simply listed equations or gave diagrams where a proof was hinted at rather than shown. In other cases a proof was shown but it was declared to be an established method after some fashion. Tang Mathematics By the Tang Dynasty study of math was fairly standard in the great schools.Wang Xiaotong was a great mathematician in the beginning of the Tang Dynasty, and he wrote a book: Jigu suanjing (Continuation of Ancient Mathematics). The table of sines by the Indian mathematician, Aryabhata, were translated into the Chinese mathematical book of the Kaiyuan Zhanjing, compiled in 718 AD during the Tang Dynasty. Although the Chinese excelled in other fields of mathematics such as solid geometry, binomial theorem, and complex algebraic formulas, early forms of trigonometry were not as widely appreciated as in the contemporary Indian and Islamic mathematics. I-Xing, the mathematician and Buddhist monk was credited for calculating the tangent table. Instead, the early Chinese used an empirical substitute known as chong cha, while practical use of plane trigonometry in using the sine, the tangent, and the secant were known.
Wang Xiaotong on Right Triangles: Six Problems from 'Continuation of Ancient Mathematics' (Seventh Century AD).
Source: East Asian Science, Technology & Medicine . 2014, Issue 37, p12-35. 24p. Author(s): Tina Su-lyn Lim; Wagner, Donald B. Abstract: Wang Xiaotong's Jigu suanjing is primarily concerned with problems in solid and plane geometry leading to polynomial equations which are to be solved numerically using a procedure similar to Horner's Method. We translate and analyze here six problems in plane geometry. In each case the solution is derived using a dissection of a 3-dimensional object. We suggest an interpretation of one fragmentary comment which at first sight appears to refer to a dissection of a 4-dimensional object. Copyright of East Asian Science, Technology & Medicine is the property of East Asian Science, Technology & Medicine and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. The history of the text is discussed by the modern editors, Qian Baocong (1963: 490–491) and Guo Shuchun and Liu Dun (1998: 1: 21–22); see also He Shaogeng (1989). All extant editions of Jigu suanjing go back to an edition printed in the early thirteenth century AD. One hand-copy of this edition survived to the twentieth century; Qian Baocong appears to have seen it.5 It is now lost, but in 1684 a copy was included in a collectaneum, Jiguge congshu 汲古閣叢書. This version, the oldest surviving text of the Jigu suanjing, is now available in facsimile on the World Wide Web. 6 The best Qing critical edition is that of Li Huang (1832); 7 others are by Dai Zhen (1777), Bao Tingbo (1780), Zhang Dunren (1803), and the Korean mathematician Nam Py?ng-Gil (1820–1869).8 All of these are available on the Web. The standard modern edition has long been Qian Baocong’s (1963, 2: 487–527; important corrections, 1966), but that of Guo Shuchun and Liu Dun (1998) has much to recommend it. In the parts of the text treated in the present article there is no important difference between the two. The 1684 edition is marred by numerous obvious scribal errors which must be corrected by reference to the mathematical context. Guo and Liu (1998, 1: 22) note that Li Huang introduced ca. 700 emendations to the text, and that Qian Baocong followed most of these but introduced 20 new emendations. The part of the text considered here gives special difficulties, for it appears that the last few pages of the Southern Song hand copy were damaged, and a great many characters are missing. The problems involved in reconstructing the text are considered further below. The Jigu suanjing has not been much studied in modern times. The two critical editions, already mentioned, do not explicitly comment on the mathematical content. Lin Yanquan (2001) translates the text into modern Chinese, expresses the calculations in modern notation, and gives derivations of some of the formulas. Deeper studies of individual parts of the text are by Shen Kangshen (1964), Qian Baocong (1966), He Shaogeng (1989), Wang Rongbin (1990), Guo Shiying (1994), and Andrea Bréard (1999: 95–99, 333–336, 353–356; 2002). The derivations of Problems 15 and 17 (Figures 2 and 3–4 below) have been reconstructed by Lin Yanquan, by He Shaogeng, and by Wang Rongbin; our reconstruction of the derivation of Problem 19 (Figure 5) is new. The comments in the text give special difficulties, for two reasons. Being written in smaller characters, they are more subject to banal scribal errors, and their content is more abstract and complex than the main text. They have hardly been studied at all by modern scholars: the most recent serious study we have found is that of Luo Tengfeng (1770–1841) [1993].
Knowledge Graph
Examples
1 Ji Gu Suan Jing was the work of early Tang dynasty calendarist and mathematician Wang Xiaotong, written some time before the year 626, when he presented his work to the Emperor.
2 Ji Gu Suan Jing contains the wisdom of Chinese people.
3 Wang Xiaotong is the author of the Jigu Suanjing