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 Sòng-Yuán 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
Reminder: AI generated list - beware of hallucinations. Corrections and additions welcome
- 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.
Links
- Companion xìcǎo: KR3fc057 Fāngchéng xīnshù cǎo, KR3fc058 Gōugǔ suànshù xìcǎo
- Sòng-Yuán precursors in the same problem domain: KR3fc015 Cèyuán hǎijìng
- CBDB (author): https://cbdb.fas.harvard.edu/cbdbapi/person.php?id=77136