The Nine Chapters on the Mathematical Art
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The Nine Chapters on the Mathematical Art (traditional Chinese: 九章算術; simplified Chinese: 九章算术; pinyin: Jiǔzhāng Suànshù) is a Chinese mathematics book, composed by several generations of scholars from the 10th-2nd century BCE, and the latest stage being the 1st century CE. This book is the second earliest surviving mathematical text from China, the first being Suàn shù shū. It lays out an approach to mathematics that centres on finding the most general methods of solving problems, which may be contrasted with the approach common to ancient Greek mathematicians, who tended to deduce propositions from an initial set of axioms.
Entries in the book usually take the form of a statement of a problem, followed by the statement of the solution, and an explanation of the procedure that led to the solution.
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[edit] History
The full title of The Nine Chapters on the Mathematical Art appears on two bronze standard measures dated to 179 AD, yet there is speculation that the same book existed beforehand only under different titles.[1]
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 Nine Chapters reached its final form. The method of chapter 7 was not found in Europe until the 13th century, and the method of chapter 8 uses Gaussian elimination before Carl Friedrich Gauss (1777–1855).[2] There is also the mathematical proof given in the treatise for the Pythagorean theorem.[3] There are also features of ancient Western mathematics that are not found in ancient China. The influence of The Nine Chapters greatly assisted the development of ancient mathematics in the regions of Korea and Japan. Its influence on mathematical thought in China persisted until the Qing Dynasty era.
Liu Hui wrote a very detailed commentary on this book in 263. He analyses the procedures of the Nine Chapters step by step, in a manner which is clearly designed to give the reader confidence that they are reliable, although he is not concerned to provide formal proofs in the Euclidean manner. Liu's commentary is of great mathematical interest in its own right. Liu credits the earlier mathematicians Zhang Cang (fl. 165 BC - d. 142 BC) and Geng Shouchang (fl. 75 BC-49 BC) (see armillary sphere) with the initial arrangement and commentary on the book, yet Han Dynasty records do not indicate the names of any authors of commentary, as they are not mentioned until the 3rd century.[4]
The Nine Chapters is an anonymous work, and its origins are not clear. Until recent years there was no substantial evidence of related mathematical writing that might have preceded it, spare mathematical work by those such as Jing Fang (78–37 BC) and Zhang Heng (78–139) and the geometry clauses of the Mozi of the 4th century BC. This is no longer the case. The Suàn shù shū (算數書) or writings on reckoning is an ancient Chinese text on mathematics approximately seven thousand characters in length, written on 190 bamboo strips. It was discovered together with other writings in 1983 when archaeologists opened a tomb at Zhangjiashan in Hubei province. From documentary evidence this tomb is known to have been closed in 186 BC, early in the Western Han dynasty. While its relationship to the Nine Chapters is still under discussion by scholars, some of its contents are clearly paralleled there. The text of the Suan shu shu is however much less systematic than the Nine Chapters; and appears to consist of a number of more or less independent short sections of text drawn from a number of sources.
[edit] Table of Contents
Contents of the Nine Chapters are as follows:
- 方田 Fang tian - Rectangular fields. Areas of fields of various shapes; manipulation of vulgar fractions.
- 粟米 Su mi - Millet and rice. Exchange of commodities at different rates; pricing.
- 衰分 Cui fen - Proportional distribution. Distribution of commodities and money at proportional rates.
- 少廣 Shao guang - The lesser breadth. Division by mixed numbers; extraction of square and cube roots; dimensions, area and volume of circle and sphere.
- 商功 Shang gong - Consultations on works. Volumes of solids of various shapes.
- 均輸 Jun shu - Equitable taxation. More advanced problems on proportion.
- 盈不足 Ying bu zu - Excess and deficit. Linear problems solved using the principle known later in the West as the rule of false position.
- 方程 Fang cheng - The rectangular array. Problems with several unknowns, solved by a principle similar to Gaussian elimination.
- 勾股 Gou gu - Base and altitude. Problems involving the principle known in the West as the Pythagorean theorem.
[edit] Accessibility
A full translation and study of the Nine Chapters and Liu Hui's commentary is available in SHEN Kangshen "The Nine Chapters on the Mathematical Art" Oxford 1999. ISBN 0-19-853936-3
A French translation with detailed scholarly addenda and a critical edition of the Chinese text of both the book and its commentary is Chemla, Karine, and Shuchun Guo. 2004. Les neuf chapitres: le classique mathématique de la Chine ancienne et ses commentaires. Paris: Dunod.
[edit] See also
[edit] Notes
[edit] References
- Needham, Joseph (1986). Science and Civilization in China: Volume 3, Mathematics and the Sciences of the Heavens and the Earth. Taipei: Caves Books, Ltd.
- Straffin, Philip D. "Liu Hui and the First Golden Age of Chinese Mathematics," Mathematics Magazine (Volume 71, Number 3, 1998): 163–181.
