Jiègēn fāngfǎ qiǎnshuō 借根方法淺說
An Elementary Exposition of the Jiè-gēn-fāng [Algebraic-Equation] Method by 劉衡 (撰)
About the work
劉衡 Liú Héng’s (1776–1841) introductory treatise in 2 juàn on the jiègēnfāng 借根方 method — the European-derived algebraic-equation procedure introduced to China by the Jesuit mathematicians in the early eighteenth century and developed as the principal Chinese-language algebraic apparatus until the mid-nineteenth-century reception of symbolic algebra under 李善蘭 Lǐ Shànlán. The title’s qiǎnshuō 淺說 (“shallow exposition” — i.e. elementary primer) indicates the work’s pedagogical orientation.
Abstract
The jiègēnfāng (literally “borrowing-of-root method”) is the Chinese transliteration of the European symbolic-algebraic procedure for setting up and solving polynomial equations. Introduced to China by the Kāng-xī-era Jesuits — particularly Tomás Pereira (徐日昇 Xú Rìshēng) and Jean-François Foucquet — and codified in the imperially-commissioned mathematical compendia Shùlǐ jīngyùn 數理精蘊 of 1722 (KR3fa008), the method introduces the unknown as a “borrowed root” gēn 根 (= radix) and constructs polynomial equations in this unknown to represent the problem conditions. The procedure is in effect Cardano-Vièta-style algebra, transliterated into Chinese terminology.
The jiègēnfāng was the dominant algebraic apparatus in eighteenth-century Chinese mathematics, in part because the indigenous SòngYuán tiānyuán 天元 method had been lost from active practice by the mid-Míng and was only recovered for active use under 戴震 Dài Zhèn’s Sìkù mathematical recovery project of the 1770s. After that recovery, the two methods coexisted in Chinese mathematical writing, with the jiègēnfāng preferred for elementary teaching and the recovered tiānyuán for theoretical work (cf. 李銳’s KR3fc060 Kāifāng shuō).
Liú Héng’s 2-juàn qiǎnshuō is an elementary primer on the jiègēnfāng. It presents the method’s basic apparatus — setting up the equation, the standard solution procedures for first-, second-, and third-degree equations, the rules for combining algebraic expressions — and a graduated sequence of worked examples. The treatise serves as a teaching counterpart to 汪曰楨 Wāng Yuēzhēn’s KR3fc061 Rújī yǐnméng on the indigenous side: where Wāng Yuēzhēn introduces the rújī / tiānyuán tradition, Liú Héng introduces the jiègēnfāng.
Dating: Liú Héng lived 1776–1841. notBefore 1800; notAfter 1841.
Translations and research
- Jami, Catherine. 2012. The Emperor’s New Mathematics: Western Learning and Imperial Authority during the Kangxi Reign (1662–1722). Oxford: Oxford University Press. — Authoritative study of the jiè-gēn-fāng tradition’s introduction under Kāng-xī.
- Tián Miǎo 田淼. 2003. Zhōng-guó shù-xué de xī-huà lì-chéng 中國數學的西化歷程. Jǐ-nán: Shān-dōng jiāo-yù chū-bǎn-shè.
- Hán Qí 韓琦. 2018. Kāng-xī huáng-dì yǔ Yē-sū-huì shì 康熙皇帝與耶穌會士. Bě-jīng: Shāng-wù yìn-shū-guǎn.