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勾股算術細草

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Text ID
KR3fc058
Date
1795–1817
Dynasty
清

Gōugǔ suànshù xìcǎo 勾股算術細草

Detailed Working of Right-Triangle Arithmetic by 李銳 (撰)

About the work

李銳 Lǐ Ruì’s (1768–1817) procedural treatise in 2 juàn on the gōugǔ 勾股 (right-triangle) problem domain — the subject matter of the Jiǔzhāng suànshù 九章算術 Chapter 9 Gōugǔ and the closely-related Zhōubì suànjīng 周髀算經 (KR3fc001’s companion classic). The title’s xìcǎo 細草 indicates the genre — worked-example exposition with step-by-step computational procedure.

Abstract

The gōugǔ (literally “leg and thigh,” i.e. the two legs of a right triangle, with the hypotenuse xián 弦 understood) is the indigenous Chinese formulation of the Pythagorean theorem and its associated problem-types: given two of {leg, leg, hypotenuse, perimeter, area, sum of two sides, difference of two sides}, find the remaining elements. The Jiǔzhāng Chapter 9 presents the basic procedures and a representative selection of problem-types; the Zhōubì suànjīng embeds the right-triangle theorem in an astronomical-cosmological framework as the basis of gàitiān 蓋天 (“heaven-as-cover”) cosmography.

Lǐ Ruì’s 2-juàn xìcǎo presents a modernised, systematic exposition of the gōugǔ problem domain. The first juàn presents the basic procedures with worked examples; the second extends to harder problem-types (involving simultaneous use of multiple gōugǔ relations, or the use of gōugǔ in combination with the area / perimeter / circular-segment apparatus). Where the Jiǔzhāng Chapter 9 presents the procedures tersely, Lǐ Ruì’s exposition supplies the intermediate steps, the algebraic derivations of the procedures from the basic Pythagorean identity, and the geometric figures clarifying the configurations.

The treatise is the central member of Lǐ Ruì’s three-work procedural-exposition sequence (KR3fc057 Fāngchéng xīnshù cǎo — linear systems; the present text — right triangles; KR3fc059 Húshǐ suànshù xìcǎo — arcs and versed-sines). Together they constitute Lǐ Ruì’s pedagogical project of making the recovered Jiǔzhāng-tradition procedures accessible to the educated mathematical reader of the JiāDào era.

Dating: composed during Lǐ Ruì’s mature productive period. notBefore 1795; notAfter 1817.

Translations and research

  • Martzloff, Jean-Claude. 1997 [2006]. A History of Chinese Mathematics. Berlin: Springer.
  • Cullen, Christopher. 1996. Astronomy and Mathematics in Ancient China: the Zhou bi suan jing. Cambridge: Cambridge University Press. — Treats the gōu-gǔ tradition in its astronomical-cosmological framework.
  • Wú Wénjùn 吳文俊, ed. 1985. Zhōng-guó shù-xué shǐ dà-xì 中國數學史大系, vol. 7.

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