Kāipíngfāng shuō 開平方說
Exposition of Square-Root Extraction by 陳藎謨 (撰)
About the work
A brief late-Míng monograph on square-root extraction by 陳藎謨 Chén Jìnmó, companion to his KR3fc033 Kāilìfāng shuō (cube-root extraction). Together the two pieces form Chén Jìnmó’s contribution to the late-Míng systematization of the kāifāng (root-extraction) tradition, parallel to 徐光啟 Xú Guāngqǐ’s KR3fc029 Dìngfǎ píngfāng suànshù.
Abstract
The work systematically expounds the kāipíngfāng 開平方 procedure for extracting square roots — the column-by-column digit-extraction method going back to the KR3f0032 Jiǔzhāng Shǎoguǎng 少廣 chapter as systematized by Liú Huī’s commentary. Where the indigenous tradition had presented the procedure schematically with practical-estimation guidance for the “next digit” determination, Chén Jìnmó’s shuō restates the procedure with explicit decision-rules and full geometric justification.
The geometric justification follows the area-decomposition argument going back to Liú Huī’s Jiǔzhāng commentary: the square root of N is found by inscribing a square of area N and successively determining its side-length by the area-relations of the partial squares-and-rectangles into which the unit-square decomposes. Chén Jìnmó presents this in the explicit Euclidean-deductive style that the late-Míng Jesuit-translation period had introduced; the diagrams in the shuō follow the Jǐhé yuánběn (Euclid II) conventions.
The work belongs to the broader late-Míng project of restating indigenous Chinese mathematical procedures in Euclidean-deductive form, alongside 徐光啟 Xú Guāngqǐ’s KR3fc029 Dìngfǎ píngfāng suànshù. The two works are conventionally read together as the principal late-Míng theoretical treatments of square-root extraction.
For Chén Jìnmó’s broader intellectual context see the Person note at 陳藎謨.
Translations and research
No substantial English-language secondary literature located on Chén Jìnmó’s root-extraction work specifically. For the general late-Míng Sino-European integration see the references at KR3fc030.
- Engelfriet, Peter M. 1998. Euclid in China. Leiden: Brill. — Provides context for the Euclidean-deductive restatement of Chinese root-extraction.
- Lam Lay-Yong. 1970. “The Geometrical Basis of the Ancient Chinese Square-Root Method.” Isis 61.1: 92–102. — Foundational treatment of the indigenous kāi-fāng procedure that Chén Jìnmó systematizes.
- Wú Wénjùn 吳文俊, ed. 1985. Zhōng-guó shù-xué shǐ dà-xì 中國數學史大系, vol. 6.