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開方說

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Text ID
KR3fc060
Date
1795–1817
Dynasty
清

Kāifāng shuō 開方說

Discourse on Root-Extraction [and Polynomial Equations] by 李銳 (撰)

About the work

李銳 Lǐ Ruì’s (1768–1817) theoretical treatise in 3 juàn on the structure of polynomial equations — the most original and theoretically-developed of his works, and one of the most important indigenous Chinese mathematical works of the entire Qīng period. Despite the conservative title (kāifāng 開方 literally “extracting the [square / cube / higher] root,” the indigenous Chinese label for the numerical solution of polynomial equations), the treatise’s actual subject is the theory of polynomial equations — the relations between the coefficients and the roots.

Abstract

The Kāifāng shuō is Lǐ Ruì’s response to a problem that had emerged in the correspondence between Lǐ Ruì and 汪萊 Wāng Lái 汪萊 (1768–1813) over the years 1800–1813. Wāng Lái’s Héngzhāi suànxué 衡齋算學 had identified a class of polynomial equations possessing multiple positive real roots — equations of a type the Chinese mathematical tradition (working with the jiègēnfāng 借根方 borrowing-of-root or with the indigenous tiānyuán 天元 celestial-element method) had typically treated as if they had a single root. Wāng Lái catalogued these “multi-root” equations and identified the algebraic conditions under which they arose.

Lǐ Ruì’s Kāifāng shuō takes Wāng Lái’s classification as starting-point and develops the systematic theory. Working from the SòngYuán algebraic apparatus (the zēngchéng kāifāng 增乘開方 procedure of 賈憲 Jiǎ Xiàn and 秦九韶 Qín Jiǔsháo, KR3fc008) and the European jiègēnfāng introduced into China by the Jesuits, the treatise systematically catalogues the relations between the number of positive real roots of a polynomial equation, the signs of its coefficients, and the sign changes in the coefficient sequence. The work in effect develops a form of Descartes’s rule of signs and a partial theory of the discriminant, derived independently from the indigenous Chinese algebraic tradition without direct knowledge of European developments.

The Kāifāng shuō is the high-water mark of late-imperial Chinese theoretical mathematics. After Wāng Lái’s death in 1813 and Lǐ Ruì’s own in 1817, the theoretical-algebraic project they had developed in correspondence did not continue. The synthesis with European mathematics was deferred to 李善蘭 Lǐ Shànlán’s mid-nineteenth-century translations and original work (KR3fc078 Zégǔxī zhāi suànxué).

Dating: composed during Lǐ Ruì’s mature productive period; the work develops directly out of the Wāng Lái – Lǐ Ruì correspondence of 1800–1813, so the latter half of that decade is the natural compositional window. notBefore 1810 (allowing for the development of the theoretical apparatus through the correspondence); notAfter 1817 (death year). Note: catalog gives only Qīng dynasty; tighter bracket adopted here on the basis of the documented Wāng – Lǐ correspondence (1800–1813) which is the prerequisite for the work’s content.

Translations and research

  • Tian, Miao [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.
  • Horng, Wann-Sheng [洪萬生]. 1991. “Li Shanlan, the Impact of Western Mathematics in China during the Late 19th Century.” Ph.D. diss., City University of New York. — Earlier chapters cover Wāng Lái and Lǐ Ruì.
  • Tian, Miao. 1999. “Jiè-gēn fāng, tiān-yuán and Kāi-fāng shuō.” Historia Mathematica 26: 60–75.
  • Wú Wénjùn 吳文俊, ed. 1985. Zhōng-guó shù-xué shǐ dà-xì 中國數學史大系, vol. 7.

Other points of interest

The Kāifāng shuō is in many ways the most theoretically ambitious work of Chinese mathematics produced under the Qīng evidential project. It demonstrates that the indigenous Chinese algebraic tradition, given sufficient theoretical development, was capable of producing results comparable in generality to those of contemporary European algebra — even when working substantially in isolation. The work is sometimes taken (e.g. by Tian Miao) as the principal counterexample to a strong “Needham-style” thesis about the necessary stagnation of Chinese mathematics in the Qīng.

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