Kāilìfāng shuō 開立方說
Exposition of Cube-Root Extraction by 陳藎謨 (撰)
About the work
陳藎謨 Chén Jìnmó’s late-Míng monograph on cube-root extraction, companion to his KR3fc032 Kāipíngfāng shuō. The work systematizes the kāilìfāng 開立方 (cube-root) procedure with the same Euclidean-deductive methodology applied to square-root extraction in the companion piece.
Abstract
The work extends the Kāipíngfāng shuō’s treatment to cube-root extraction, presenting the column-by-column cube-root procedure with explicit decision-rules and full geometric justification. The geometric foundation is the cube-decomposition analogous to the square-decomposition for square-roots: the cube of side a+b decomposes into the sub-cube of side a, plus 3 rectangular boxes of dimensions a×a×b, plus 3 rectangular boxes of dimensions a×b×b, plus the sub-cube of side b — i.e., (a+b)³ = a³ + 3a²b + 3ab² + b³. The cube-root procedure works by successively determining each digit of the root by examining the residual after subtracting the lower-order contributions; the geometric decomposition justifies the precise form of the subtraction at each stage.
Chén Jìnmó’s presentation follows the indigenous Liú Huī tradition (where the same geometric decomposition appears in the Jiǔzhāng Shānggōng 商功 / Shǎoguǎng 少廣 commentary) but restates it in Euclidean-deductive form, with diagrams showing the cube-decomposition and explicit propositional justifications. The work is one of the earliest Chinese mathematical treatises to present cube-root extraction in fully demonstrative form, presaging the systematic treatments by 王孝通 Wáng Xiàotōng’s Qīng expositors (Lǐ Huáng’s KR3fc005 Jígǔ suànjīng kǎozhù, Zhāng Dūnrén’s KR3fc006 xìcǎo).
The work also discusses the connection between cube-root extraction and the solution of cubic equations — anticipating in elementary form the systematic treatment by 焦循 Jiāo Xún’s KR3fc048 Kāifāng tōngshì. Although Chén Jìnmó’s treatment remains within the procedural-cube-root framework rather than developing a fully algebraic cubic-equation method, his analysis shows substantive awareness of the underlying mathematical relationship.
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 cube-root work specifically. For general context see the references at KR3fc030 and KR3fc032.
- Lam Lay-Yong. 1970. “The Geometrical Basis of the Ancient Chinese Square-Root Method.” Isis 61.1: 92–102. — The companion treatment for square-root; the same approach applies to cube-root.
- Engelfriet, Peter M. 1998. Euclid in China. Leiden: Brill.
- Wú Wénjùn 吳文俊, ed. 1985. Zhōng-guó shù-xué shǐ dà-xì 中國數學史大系, vol. 6.