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定法平方算術

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Text ID
KR3fc029
Date
1606–1633
Dynasty
明

Dìngfǎ píngfāng suànshù 定法平方算術

Mathematical Method for Fixed-Procedure Square-Root Extraction by 徐光啟 (撰)

About the work

A brief late-Míng treatise on square-root extraction by 徐光啟 Xú Guāngqǐ (1562–1633), companion to his better-known KR3fc026 Gōugǔ yì. The work systematizes the píngfāng 平方 (square-root) extraction procedure into a dìngfǎ 定法 (“fixed method”) — a standardized step-by-step procedure with clearly-stated rules at each stage.

Abstract

The Dìngfǎ píngfāng suànshù belongs to Xú Guāngqǐ’s project of bringing the indigenous Chinese mathematical procedures into dialogue with the newly-translated Euclidean geometry. The traditional Chinese kāipíngfāng 開平方 procedure for square-root extraction — going back to the KR3f0032 Jiǔzhāng Shǎoguǎng 少廣 chapter — proceeds by a column-by-column extraction analogous to long division, with the key step being the determination of the next digit by examining the residual after each round. This procedure was stated informally in the indigenous tradition, with the “next-digit” determination treated as a matter of practical estimation rather than as a determinate rule.

Xú Guāngqǐ’s dìngfǎ contribution is to convert this procedure into a fully determinate algorithm with explicit decision-rules at each step — recasting the procedure in the style of a Euclidean-deductive systematization. The work is one of the earliest examples of Chinese mathematical writing in which a traditional procedure is restated in deductive-axiomatic form, presaging the systematic Euclidean reformulation of Chinese mathematical practice that would mature in 梅文鼎 Méi Wéndǐng and the Kāng-xī-era Méngyǎng zhāi mathematicians.

The work also provides the geometric justification for the procedure: Xú Guāngqǐ shows, by area-decomposition diagrams analogous to those of Jǐhé yuánběn II.4 (the geometric identity (a+b)² = a² + 2ab + b²), why the column-by-column procedure works. This is the same area-decomposition argument that Liú Huī had given for the procedure in his Jiǔzhāng commentary, but Xú Guāngqǐ presents it in explicit Euclidean-style demonstrative form.

Dating: the work is conventionally placed in Xú Guāngqǐ’s mature period after the 1607 Jǐhé yuánběn and before his calendar-reform appointment in 1629; notBefore 1606 (concurrent with the early-stage drafting of the Gōugǔ yì); notAfter 1633 (his death year).

For Xú Guāngqǐ’s broader biographical and intellectual context see the Person note at 徐光啟.

Translations and research

  • Engelfriet, Peter M. 1998. Euclid in China. Leiden: Brill. — Treats Xú Guāngqǐ’s broader project of bringing Chinese mathematical procedures into Euclidean-deductive form.
  • Jami, Catherine. 2011. The Emperor’s New Mathematics. Oxford: Oxford University Press.
  • Hashimoto Keizō 橋本敬造. 1988. Hsü Kuang-ch’i and Astronomical Reform. Osaka: Kansai University Press.
  • Wú Wénjùn 吳文俊, ed. 1985. Zhōng-guó shù-xué shǐ dà-xì 中國數學史大系, vol. 6.

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