The Nine Chapters on the Mathematical Art

The Nine Chapters on the Mathematical Art (九章算術) is a Chinese mathematics book, probably composed in the 1st century AD, but perhaps as early as 200 BC. This book is the earliest surviving mathematical text from China that has come down to us by being copied by scribes and (centuries later) printed. It reveals an approach to mathematics that centres on finding the most general methods of solving problems rather than on deducing propositions from an initial set of axioms, in the manner found amongst leading ancient Greek mathematicians.
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.
Contents of the Nine Chapters are as follows
 Fang tian  Rectangular fields Areas of fields of various shapes; manipulation of vulgar fractions.
 Su mi  Millet and rice Exchange of commodities at different rates; pricing.
 Cui fen  Proportional distribution Distribution of commodities and money at proportional rates.
 Shao guang  The lesser breadth Division by mixed numbers; extraction of square and cube roots; dimensions, area and volume of circle and sphere.
 Shang gong  Consultations on works Volumes of solids of various shapes.
 Jun shu  Equitable taxation More advanced problems on proportion.
 Ying bu zu  Excess and deficit Linear problems solved using the principle known later in the West as the rule of false position.
 Fang cheng  The rectangular array Problems with several unknowns, solved by a principle similar to Gaussian elimination.
 Gou gu  Base and altitude Problems involving the principle known in the West as the Pythagorean theorem.
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. There is therefore little historical value in speculations about who was "more advanced" at any given period. Nevertheless we may note that the method of chapter 7 was not found in Europe until the 13th century, and the method of chapter 8 is not found before the sixteenth century. Of course there are also features of ancient Western mathematics that are not found in ancient China.
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.
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. This is no longer the case. The Suan shu shu 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.
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 0198539363zh:九章算术 ja:九章算術