Kāifāng tōngshì 開方通釋
General Exposition of Root Extraction and Equation Solving by 焦循 (撰)
About the work
焦循 Jiāo Xún’s monograph on root extraction and equation-solving in 2 juàn, the closing volume in the collected Lǐtáng xuésuàn jì (KR3fc042). The work systematizes the kāifāng 開方 method — the general algorithm for solving polynomial equations of arbitrary degree.
Abstract
The kāifāng method, in its developed SòngYuán form, is essentially what is now called the Horner-method for polynomial root extraction: an iterative algorithm that, given a polynomial P(x) and an initial estimate for a root, produces successively more refined approximations by a procedure equivalent to repeated polynomial-long-division. The method had been systematized by 秦九韶 Qín Jiǔsháo (the zēngchéng kāifāng 增乘開方 method) and applied to high-degree equations by 朱世傑 Zhū Shìjié; it was forgotten in the late Míng (parallel to the forgetting of tiānyuányī), recovered in the QiánJiā era through Méi Juéchéng’s identification with the European jiègēnfāng and Dài Zhèn’s Sìkù-period recovery of the SòngYuán sources.
Jiāo Xún’s Kāifāng tōngshì — together with 李銳 Lǐ Ruì’s KR3fc060 Kāifāng shuō — is the principal Yáng-zhōu-circle systematic statement of the recovered method. The two juàn cover:
(1) The basic algorithm for square-and-cube-root extraction (the special case where the equation is x² = N or x³ = N), with full procedural detail and geometric justification. Treatment of the zhèng (positive-root) and fù (negative-root) cases.
(2) The general algorithm for polynomial-equation solution by zēngchéng kāifāng. Treatment of the cases of multiple roots, of irrational roots (where the algorithm produces successive rational approximations), and of the relations between the coefficients and the root-structure (early statements of what would later be called Vieta’s formulas).
The work is one of the principal Qián-Jiā-era theoretical-mathematical productions. Its systematic engagement with the general theory of polynomial equations — going beyond the practical-computational kāifāng tradition into substantive algebraic theory — anticipates the more developed 李銳 Lǐ Ruì equation-theory works (KR3fc060) and the late-Qīng synthesis with European algebraic theory (in 李善蘭 Lǐ Shànlán’s KR3fc078 Zégǔxī zhāi suànxué).
For Jiāo Xún’s broader project see KR3fc042.
Dating: bracketed by Jiāo Xún’s productive period 1797–1820.
Translations and research
- Lam Lay-Yong. 1980. “The Chinese Connection between the Pascal Triangle and the Solution of Numerical Equations of any Degree.” Historia Mathematica 7: 407–424. — Treats the Horner-method recovery in the Qián-Jiā era.
- Hé Bǐngyù 何丙郁 (John Hoe). 1977. Les systèmes d’équations polynomes dans le Siyuan yujian (1303). Paris: Collège de France.
- 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.