勾股義

Gōugǔ yì 勾股義

Meaning of the Right Triangle by 徐光啟 (撰)

About the work

A brief late-Míng treatise on the right triangle in 1 juàn by 徐光啟 Xú Guāngqǐ (1562–1633), composed in the immediate aftermath of his collaboration with Matteo Ricci on the partial translation of Euclid’s Elements as Jǐhé yuánběn 幾何原本 (1607). The catalog records “extent: 1562-1633” — interpretable as Xú Guāngqǐ’s lifedates being inadvertently entered as the extent field. CBDB id 30598 confirms the dates.

Abstract

The Gōugǔ yì belongs to Xú Guāngqǐ’s project of placing the indigenous Chinese mathematical tradition into dialogue with the newly-translated Euclidean geometry. Where 顧應祥 Gù Yìngxiáng’s KR3fc025 Gōugǔ suànshù had treated the right triangle as a stand-alone procedural-mathematical topic to be worked through problem-by-problem, Xú Guāngqǐ’s Gōugǔ yì — composed within months of the Jǐhé yuánběn’s publication — treats the right triangle as a topic to be demonstrated (the yì 義 of the title — “meaning” or “principle” — corresponds to the demonstrative-deductive Euclidean λόγος / ratio).

The work systematizes the metric relations of the right triangle by deriving them from Euclidean propositions translated in the Jǐhé yuánběn (notably the propositions of Books I and VI on similar triangles and the Pythagorean theorem itself, Book I proposition 47). Xú Guāngqǐ presents fifteen propositions on the right triangle, each accompanied by a Euclidean-style demonstration referring back to the Jǐhé yuánběn by proposition-number. The result is the first systematic application of the deductive-demonstrative method to the indigenous Chinese gōugǔ tradition — an explicit programmatic intervention in the indigenous tradition by the methodology of the newly-imported Euclidean geometry.

The work is one of the most consequential mathematical documents of the late Míng. It establishes a methodological pattern — deriving indigenous results from Euclidean foundations — that would shape Chinese mathematical writing through the QiánJiā era (梅文鼎 Méi Wéndǐng’s KR3f0049 Gōugǔ jǔyú and Lǐ Shànlán’s mid-19th-century mathematical synthesis are direct descendants of Xú Guāngqǐ’s approach). The work also exemplifies the bǔrú 補儒 (“supplementing Confucianism”) rhetoric of late-Míng Jesuit-collaborator scholars: where Gù Yìngxiáng’s Cèyuán hǎijìng exposition had been forced to omit the lost tiānyuányī method that gave the work its substantive coherence, Xú Guāngqǐ’s Gōugǔ yì offers Euclidean demonstration as a substitute for the lost indigenous foundation.

Dating: the work is conventionally dated 1606–1608 in the immediate aftermath of the Jǐhé yuánběn (1607) — notBefore 1606, allowing for early-stage drafting concurrent with the Jǐhé translation; notAfter 1608, by which time the work was in circulation.

Translations and research

• Engelfriet, Peter M. 1998. Euclid in China: The Genesis of the First Chinese Translation of Euclid’s Elements, Books I–VI (Jihe yuanben; Beijing, 1607) and Its Reception up to 1723. Leiden: Brill. — The standard Western-language treatment; includes substantial discussion of the Gōu-gǔ yì as Xú Guāngqǐ’s principal independent mathematical work after the Jǐ-hé yuán-běn.
• Jami, Catherine. 2011. The Emperor’s New Mathematics: Western Learning and Imperial Authority during the Kangxi Reign (1662–1722). Oxford: Oxford University Press. — Provides the broader context of late-Míng / early-Qīng mathematical encounter.
• Bréard, Andrea. 2019. Nine Chapters on Mathematical Modernity. Cham: Springer.
• Wú Wénjùn 吳文俊, ed. 1985. Zhōng-guó shù-xué shǐ dà-xì 中國數學史大系, vol. 6.

Links

• Foundational source: Jǐhé yuánběn 幾何原本 (the Ricci/Xú Euclid translation)
• Late-Míng counterpoint: KR3fc025 Gōugǔ suànshù (Gù Yìngxiáng)
• Successor tradition: KR3f0049 Gōugǔ jǔyú (Méi Wéndǐng), KR3fc038 Gōugǔ gēyuán jì (Dài Zhèn)
• CBDB: https://cbdb.fas.harvard.edu/cbdbapi/person.php?id=30598