Tiānyuányī shì 天元一釋
Exposition of the Heavenly-Origin-as-One (Algebraic-Equation) Method by 焦循 (撰)
About the work
The principal tiānyuányī exposition in 焦循 Jiāo Xún’s collected Lǐtáng xuésuàn jì (KR3fc042), in 3 juàn. The most theoretically-developed Qián-Jiā-era treatment of the SòngYuán algebraic-equation method.
Abstract
The tiānyuányī 立天元一 method — the systematic polynomial-equation notation developed in the SòngYuán mathematical tradition by 秦九韶 Qín Jiǔsháo and 李冶 Lǐ Yě (see KR3f0042 Cèyuán hǎijìng and KR3f0041 Shùxué jiǔzhāng) — had been forgotten in China by the late Míng (see KR3f0043 Cèyuán hǎijìng fēnlèi shìshù for the documentation of the forgetting). Recovery began with Méi Juéchéng 梅㲄成’s early-eighteenth-century identification of tiānyuányī with the European jiègēnfāng (borrowed-root) method imported under the Kāngxī court; it accelerated with 戴震 Dài Zhèn’s Sìkù-period recovery of the Cèyuán hǎijìng and other SòngYuán sources. Jiāo Xún’s Tiānyuányī shì is the principal Yáng-zhōu-circle systematic exposition of the recovered method.
The three juàn:
(1) The basic tiānyuányī notation: representing polynomial expressions as columns of counting-rod numerals with the yuán 元 marker indicating the variable position; constructing equations by setting two such columns equal; the basic operations of addition and subtraction on the rod-columns.
(2) Polynomial multiplication, the elimination of variables, and the reduction to a single-variable equation. Includes systematic comparison with the jiègēnfāng method, showing the precise correspondence between the two notational systems and the equivalence of the underlying algorithms.
(3) The solution of the resulting single-variable polynomial equation by kāifāng (root-extraction) methods, with detailed worked examples drawn from 李冶 Lǐ Yě’s Cèyuán hǎijìng problems.
The work is the principal Qián-Jiā-era systematic statement of the indigenous Chinese algebra. Together with 李銳 Lǐ Ruì’s parallel equation-theory works, it provides the working mid-Qīng mathematician with a recovered procedural-and-theoretical access to the SòngYuán algebraic tradition that had been inaccessible for some five centuries. The methodological commitment to integrating indigenous and imported notations — present throughout Jiāo Xún’s project — makes this work also a foundational document of the QiánJiā integrative Xīfǎ Zhōngyuán program.
For Jiāo Xún’s broader project see KR3fc042.
Dating: bracketed by Jiāo Xún’s productive period 1797–1820.
Translations and research
- Hé Bǐngyù 何丙郁 (John Hoe). 1977. Les systèmes d’équations polynomes dans le Siyuan yujian (1303). Paris: Collège de France. — Treats Jiāo Xún’s tiān-yuán-yī exposition as a key document of the Qián-Jiā recovery.
- Bréard, Andrea. 1999. Re-Kreation eines mathematischen Konzepts im chinesischen Diskurs. Stuttgart: Steiner.
- Wú Wénjùn 吳文俊, ed. 1985. Zhōng-guó shù-xué shǐ dà-xì 中國數學史大系, vol. 7.
- Lǐ Zhàohuá 李兆華. 2001. “Jiāo Xún de shù-xué chéngjiù.” Zhōng-guó shǐ yán-jiū.