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方程新術草

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Text ID
KR3fc057
Date
1795–1817
Dynasty
清

Fāngchéng xīnshù cǎo 方程新術草

Detailed Working of a New Method for the Fāng-chéng [Linear Systems] by 李銳 (撰)

About the work

李銳 Lǐ Ruì’s (1768–1817) mathematical treatise in 1 juàn presenting a new procedure (xīnshù 新術) for the solution of simultaneous linear equations — the problem domain of the Fāngchéng 方程 chapter (Chapter 8) of the Jiǔzhāng suànshù 九章算術 (KR3fc001). The title’s cǎo 草 (“detailed working”) indicates that the treatise presents the procedure step-by-step with worked numerical examples, following the xìcǎo 細草 genre conventions developed by 楊輝 Yáng Huī and refined under 戴震 Dài Zhèn’s Sìkù mathematical recovery project.

Abstract

The Fāngchéng problem — solution of n simultaneous linear equations in n unknowns — is one of the principal computational problems of the indigenous Chinese mathematical tradition. The Jiǔzhāng suànshù presents an elimination procedure on the counting-board that is effectively a Gauss-Jordan elimination over the rationals (the Fāngchéng shù 方程術 of the Jiǔzhāng Chapter 8). The procedure is fully general but computationally tedious: the standard Jiǔzhāng examples involve 3- or 4-variable systems, and the labour of solution grows roughly as the cube of the system size.

Lǐ Ruì’s xīnshù presents an improved procedure that organises the elimination more efficiently for larger systems and shows worked examples through to systems of higher order. The treatise also addresses the zhèngfù 正負 (positive-negative) sign-bookkeeping that becomes nontrivial when the elimination produces negative intermediate coefficients — extending the procedural account of negative numbers given in the Jiǔzhāng Fāngchéng commentary by 劉徽 Liú Huī.

The work is one of three procedural-exposition treatises Lǐ Ruì produced (the others being KR3fc058 Gōugǔ suànshù xìcǎo and KR3fc059 Húshǐ suànshù xìcǎo) presenting modernised expositions of three central Jiǔzhāng topics — linear systems (Chapter 8), right triangles (Chapter 9), and arc-segment / versed-sine geometry (Chapter 9 extensions). The three treatises together form Lǐ Ruì’s contribution to the QiánJiā project of recovering and modernising the procedural content of the Jiǔzhāng tradition.

Dating: composed during Lǐ Ruì’s mature productive period in 阮元 Ruǎn Yuán’s patronage circle. notBefore 1795; notAfter 1817. The work circulated in 阮元 Ruǎn Yuán’s mathematical circle and was discussed in the 汪萊 Wāng Lái – Lǐ Ruì correspondence of 1800–1813.

Translations and research

  • Martzloff, Jean-Claude. 1997 [2006]. A History of Chinese Mathematics. Berlin: Springer. — Discusses Lǐ Ruì’s Fāng-chéng work in the context of the indigenous-tradition recovery.
  • Hart, Roger. 2010. The Chinese Roots of Linear Algebra. Baltimore: Johns Hopkins University Press.
  • 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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