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弧矢算術細草

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Text ID
KR3fc059
Date
1795–1817
Dynasty
清

Húshǐ suànshù xìcǎo 弧矢算術細草

Detailed Working of Arc-and-Versed-Sine Arithmetic by 李銳 (撰)

About the work

李銳 Lǐ Ruì’s (1768–1817) procedural treatise in 1 juàn on the húshǐ 弧矢 problem domain — the geometry of arc-segments of a circle, with hú 弧 “arc” and shǐ 矢 “arrow” (= versed-sine, i.e. the sagitta or the radial distance from the chord to the arc). The title’s xìcǎo 細草 again indicates the worked-example exposition genre.

Abstract

The húshǐ problem-type — given two of {arc-length, chord-length, sagitta, area-of-segment}, find the rest — is the Chinese mathematical-tradition counterpart of the European theory of circular segments. The classical Chinese treatment uses an approximate formula involving the chord, the sagitta, and the circle radius (the Jiǔzhāng Chapter 1 Fāngtián 方田 Wǎntián 宛田 procedure: S = (chord × sagitta + sagitta²) / 2), which is approximate but adequate for typical surveying applications.

The húshǐ problem was developed substantially in the SòngYuán period under 沈括 Shěn Kuò’s Mèngxī bǐtán 夢溪筆談 (with the huìyuán shù 會圓術 chord-arc procedure), 郭守敬 Guō Shǒujìng’s Shòushí 授時 calendrical apparatus (which uses arc-segment computations in the spherical-astronomy procedures), and 李冶 Lǐ Yě’s algebraic treatment in the Cèyuán hǎijìng 測圓海鏡 (KR3fc015). Lǐ Ruì’s 1-juàn xìcǎo presents the consolidated Chinese-tradition treatment, with worked examples, in modernised exposition.

The treatise completes Lǐ Ruì’s three-work procedural-exposition sequence (KR3fc057–KR3fc059). The three together — linear systems, right triangles, arc-segments — cover the principal procedural-geometric problem-types of the indigenous Jiǔzhāng tradition.

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. — Treats the hú-shǐ tradition.
  • Sivin, Nathan. 2009. Granting the Seasons. New York: Springer. — Treats Guō Shǒujìng’s arc-segment work.
  • 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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