Jiu Zhang Suanshu

from Wikipedia, the free encyclopedia

The Jiǔ Zhāng Suànshù ( Chinese  九章 算術  /  九章 算术  - "Nine Chapters of Arithmetic Art", also: 'Nine Books of Arithmetic Technique') is one of the oldest surviving Chinese mathematics books and at the same time one of the best known and most important of the Ten Mathematical Classics Collection ( Suanjing shi shu ) . The book probably shows the state of Chinese mathematics in the 1st century AD; it was also used in neighboring countries such as Korea, Japan, and Vietnam.

One page of the book


The book is a collection of 246 problems, which are grouped in nine chapters according to fields of application in everyday life (such as construction engineering, taxes, trade, land surveying, lending money). The problem, the solution and the solution are given.

  • Chapter 1: Fāng tián ( 方 田 , fangtian  - "square field"), with 38 exercises on area measurements from land surveying
  • Chapter 2: Sù mĭ ( 粟米  - “Millet and Rice”), with 46 problems from the trading environment (exchange of goods and money). Ratio calculations and fractions are practiced.
  • Chapter 3: Shuāi fēn ( 衰 分  - "Decreasing proportions") with 20 tasks, again of the type of relative tasks
  • Chapter 4: Shăo guăng ( 少 廣  /  少 广  - "Missing width"). 24 tasks about division (formulated as the task of finding the second side length of a surface, if a side length and area measure are given), pulling square and cube roots. In Liu Hui's commentary, the problem of determining the volume of a sphere is discussed, but this remains open.
  • Chapter 5: Shāng gōng ( 商 功  - "Estimation of the work"), 28 tasks from construction technology (such as canal and dike construction) on the volume of different bodies. In the commentary, Liu Hui gives the volume of the pyramid.
  • Chapter 6: Jūnshū ( 均 輸  /  均 输  - “Just distribution of goods”), 28 more advanced tasks relating to the just distribution of goods , for example tasks relating to the distribution of soldiers to garrisons at the borders, sharing and transport of goods.
  • Chapter 7: Yíng bùzú ( 盈 不足  - “Abundance and Deficiency”), 20 problems on linear equations that are solved in a trial-and-error process that corresponds to the Regula falsi , which was only known in Europe in the 13th century was.
  • Chapter 8: Fāngchéng ( 方程  /  方程  - " Rectangular arrangements "). 18 exercises on systems of linear equations that are solved with Gaussian Elimination , which was first made known in Europe by Carl Friedrich Gauß .
  • Chapter 9: Gōugŭ ( 勾股 , gōugŭ  - "right triangles"). 24 Problems About Right Triangles. Here is the Pythagorean theorem ( Gōugŭ rule in the Chinese ).


The oldest known edition of the first five chapters of the nine chapters goes back to the 13th century AD, the other known complete editions are based on reconstructions of the text at the end of the 18th century, based on quotations from the nine chapters in a today almost completely lost encyclopedia from the beginning of the 15th century. 1983–1984 inscriptions were made on bamboo strips from the period 187 to 157 BC. Discovered, whose text Suan Shu Shu is very similar to the elementary part of today's text of the Nine Chapters . Likewise, parts of the nine chapters can be dated to the early Han dynasty (206 BC - 9 AD) by means of dimensions and designations such as location and tax information used in the text . Very similar text fragments on wooden strips, which were found in the years 1899 and 1930, speak for this dating. In China at that time, the Seleucid Empire (312–63 BC) also seems to have been known and there are similarities between certain problems in the Nine Chapters and those of Babylonian mathematics, so parts of the Nine Chapters may have been influenced by Babylon. However, there are also parts of the text, e.g. B. Chapters 4 and 8, which offer no clue for a dating.

Of the many commentaries on the Nine Chapters , only those of Liu Hui (3rd century AD), Li Chunfeng (602–670 AD) and a fragment of Zu Xuan are from the 1st millennium AD (6th century AD) preserved. Liu Hui wrote in his foreword that the first Chinese emperor Qin Shihuangdi (he ruled as emperor from 221 to 210 BC) had written records burned, which destroyed ancient knowledge. Later, Zhang Cang, the governor of Beijing (around 165–142 BC), and his colleagues made a new, complete reproduction, with some parts also being updated. In addition, Geng Shouchang, second minister of agriculture (around 75–49 BC), is said to have revised certain parts.

It is now believed that the work received its present form in the 1st century AD and that new knowledge has only been included in the commentaries since then. The nine chapters of arithmetic are an important part of the collection of the ten mathematical classics, first compiled and commented by Li Chunfeng in AD 644–648 .

Interpretation of the Pythagorean theorem in the nine chapters according to Chemla

The following images show a diagram from the nine chapters and its interpretation to prove the Pythagorean theorem based on Karine Chemla's critical edition.


German edition:

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:

French / Chinese edition, critical edition:

  • Karine Chemla , Shuchun Guo: Les neuf chapitres: le classique mathématique de la Chine ancienne et ses commentaires. Dunod, Paris 2004, ISBN 2-10-049589-5 .


  • Karine Chemla: Jiuzhang Suanshu , in: Helaine Selin (Hrsg.): Encyclopaedia of the history of science, technology and medicine in non-western countries, Kluwer 2008
  • Helmuth Gericke : Mathematics in Antiquity and the Orient. Springer, Berlin a. a. 1984, pp. 172-180.
  • Jean-Claude Martzloff : A History of Chinese Mathematics. Springer, Berlin a. a. 1997, pp. 127-136.

Web links


  1. Martzloff, p. 128.
  2. Martzloff, p. 131.
  3. In the right triangle, the short cathete is called gōu ("base" or "foot") and the long cathete is called ("leg").
  4. Martzloff, p. 128 f.
  5. Martzloff, p. 129.
  6. Martzloff, p. 130 f.
  7. Martzloff, pp. 94-96.
  8. Martzloff, p. 131.
  9. Martzloff, p. 135.
  10. Martzloff, p. 129.
  11. Gericke, p. 173; Martzloff, p. 128.
  12. Martzloff, pp. 123-126.