The Nine Chapters on the Mathematical Art

The jiǔ SuanShu Zhang (Chinese九章 算术/九章 算术, Chapter Nine of arithmetic ', also ,' Nine Books arithmetic technique ') is one of the oldest extant Chinese mathematics books and one of the most famous and important collection of Ten Mathematical Classics ( Suanjing shi shu ). The book probably shows the state of Chinese mathematics of the 1st century AD, it was also used in neighboring countries such as Korea, Japan and Vietnam.

Content

The book is a collection of exercises of 246 problems, which are grouped into nine chapters corresponding fields of application in everyday life ( such as building technology, taxation, trade, land surveying, lending money ). There the problem, the solution and solving the problem is given in each case.

  • Chapter 1: Fangtian (Chinese方 田/方 田, square box '), with 38 tasks on surface measurements from the land surveying
  • Chapter 2: Sù MI (Chinese粟米/粟米, millet and rice '), with 46 problems from the field of trade ( exchange of goods and money exchange). Be practiced ratio calculations and fraction calculations.
  • Chapter 3: Shuai fēn (Chinese衰 分/衰 分, Decreasing Shares ' ) with 20 tasks, again on the type of relationship tasks
  • Chapter 4: Shao Guang (Chinese少 广/少 广, Missing width '). 24 tasks Division ( formulated as a task to find the second side length of a surface when a side length and area measure are given ), drawing of square and cube root. In the commentary of Liu Hui, the problem is discussed to determine the volume of a sphere, but remains open.
  • Chapter 5: ShangGong (Chinese商 功/商 功, estimation of labor ' ), 28 items from the building construction ( such as channel - and dike construction ) over the volume of different bodies. In the commentary Liu Hui gives the volume of the pyramid.
  • Chapter 6: Jūnshū (Chinese均 输/均 输, Equitable distribution of goods ' ), 28 advanced tasks relative to equitable distribution of goods, for example tasks to deploy soldiers to garrisons on the borders, parts and transportation of goods.
  • Chapter 7: Ying bù zú (Chinese盈 不足/盈 不足, abundance and lack ' ), 20 tasks via linear equations, which are solved in a trial-and- error method, which corresponds to the Regula falsi, the only in Europe was known 13th century.
  • Chapter 8: Fangcheng (Chinese方程/方程, rectangular arrays '). 18 tasks on systems of linear equations that are solved using Gaussian elimination, which was only known by Carl Friedrich Gauss in Europe.
  • Chapter 9: Gougu (Chinese勾股/勾股, right-angled triangles '). 24 problems about right triangles. Here we find the Pythagorean theorem (for the Chinese Gougu rule).

Tradition

The oldest, now known edition of the first five chapters of the Nine chapter goes back to the 13th century AD, the other known total expenditure based on reconstructions of the text at the end of the 18th century, the quotations from the nine chapters into one today almost completely lost Encyclopedia were created from the beginning of the 15th century. 1983-1984 inscriptions on bamboo strips from the period 187-157 BC discovered, the text of Suan Shu Shu is very similar to the elementary part of the present text of the nine chapters. Also can be used in the text by dimensions and designations such as local tax and information parts of the nine chapters in the early Han Dynasty ( 206 BC -9 AD) dating. Also very similar fragments of text speak on wooden strips, which were found in 1899 and 1930 for this dating. In China this time, also seems to have the Seleucid Empire ( 312-63 BC), known and there are similarities of certain problems of the nine chapters on such Babylonian mathematics, so parts of the nine chapters could have been Babylonian influences. However, there are also parts of the text, such as Chapters 4 and 8, which provide no evidence for a dating.

From the many comments on the nine chapters of the 1st millennium AD only by Liu Hui ( third century AD), Li Chunfeng ( 602-670 AD) and a fragment of about Xuan ( 6th century AD ) were obtained. Liu Hui wrote in his preface that the first Chinese Emperor Qin Shi Huang (who ruled as Emperor of 221-210 BC) who burned written records, which old knowledge had been destroyed. Later 've Zhang Cang, the governor of Beijing ( BC to 165-142 ), made ​​with employees a new, complete play, while some parts were also updated. In addition, to have revised ( to 75-49 BC ), specific parts Geng Shouchang, Second Minister for Agriculture.

Today it is assumed that the work was given its present form in the 1st century AD and that since received new knowledge only in the comments. The nine chapters of arithmetic are an important part of the first of Li Chunfeng 644-648 AD compiled and provided with comments Collection of the Ten Mathematical Classics.

Interpretation of the Pythagorean theorem in the nine chapters by Chemla

The pictures below give a diagram of the nine chapters again and its interpretation to the proof of the theorem of Pythagoras on the critical issue of Karine Chemla.

Interpretation by Chemla, squares of the side lengths of 3.4 on the other sides of the triangle, for example, 1

Interpretation by Chemla, sequel, The large square, according to the hypotenuse, has the same area as the sum of the Kathetenquadrate.

Expenditure

German edition:

  • Kurt Vogel: Nine books Arithmetic technology. Vieweg 1968.

English editions:

  • Kansheng Shen, et al ( eds ): The Nine Chapters on the Mathematical Art Oxford University Press, Oxford 1999, ISBN 0-19-853936-3.
  • Abridged translations published by: Florian Cajori: Arithmetic in Nine Sections. In 1893.
  • Lam Lay Yong: Jiu Zhang SuanShu, An Overview. Archive for History of Exact Science, Volume 47, 1994, pp. 1-51.

French / Chinese edition, critical edition:

  • Karine Chemla, Shuchun Guo: Les neuf Chapitres: le classique de la Chine ancienne mathématique et ses commentaires. Dunod, Paris 2004, ISBN 2-10-049589-5.
The Ten Computational Canons Book on Numbers and Computation Helmuth Gericke
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