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勾股割圓記

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Text ID
KR3fc038
Date
1755–1773
Dynasty
清

Gōugǔ gēyuán jì 勾股割圓記

Record of the Right-Triangle and Circle-Cutting by 戴震 (撰)

About the work

戴震 Dài Zhèn’s (1724–1777) treatise on the right-triangle and circle-cutting methods, in 3 juàn (the present catalog records 4 juàn, likely including an additional supplementary piece). Composed in the middle decades of Dài Zhèn’s productive period, before his appointment to the Sìkù quánshū editorial board in 1773.

Abstract

The Gōugǔ gēyuán jì is one of Dài Zhèn’s principal mathematical works and a key document of mid-Qián-lóng evidential mathematical scholarship. Dài Zhèn, although primarily known as the foundational figure of the Huī 徽-school philological kǎozhèng (evidential-research) movement, was also the principal mathematical scholar of his generation; his recovery of the KR3f0032 Jiǔzhāng suànshù and KR3f0035 Hǎidǎo suànjīng from the Yǒnglè dàdiǎn during his Sìkù service made the early classical Chinese mathematical tradition newly accessible to the QiánJiā mathematical philologists (李潢 Lǐ Huáng, 張敦仁 Zhāng Dūnrén, 焦循 Jiāo Xún, 李銳 Lǐ Ruì).

The Gōugǔ gēyuán jì belongs to Dài Zhèn’s broader project of placing the indigenous Chinese mathematical methods on a rigorous theoretical foundation. The work systematizes:

(1) The right-triangle (gōugǔ) metric relations, with explicit demonstration following the 徐光啟 Xú Guāngqǐ Gōugǔ yì (KR3fc026) Euclidean-style methodology — i.e., deriving each relation by appeal to specific propositions of the Jǐhé yuánběn.

(2) The circle-cutting (gēyuán 割圓) method for computing π, going back to 劉徽 Liú Huī’s 192-gon and Zǔ Chōngzhī 祖冲之’s 3.1415926. Dài Zhèn provides the full procedural details of the successive-polygon-doubling method and tabulates the resulting π approximations.

(3) The relation between the circle-cutting method and the chord-and-arc (xiánshǐ 弦矢) functions — i.e., the elementary trigonometric functions in their indigenous Chinese form. This sets up the broader context within which the 安圖 Mínggatu Gēyuán mìlǜ jiéfǎ (KR3fc037) infinite-series methods would be received in late-Qián-lóng Chinese mathematical practice.

Methodologically, the work shows Dài Zhèn’s characteristic concern for rigorous demonstration and explicit procedural detail. Unlike many of his contemporaries who treated mathematical procedures as practical recipes to be applied without further justification, Dài Zhèn insists on the demonstrative foundations of each procedure — a methodological posture that he would extend to his philological-classical work as well.

The work was widely cited in the late-Qián-lóng / Jiāqìng mathematical literature and shaped the Jiā-Dào-era Yángzhōu mathematical circle’s approach to circle-cutting and trigonometry, particularly 焦循 Jiāo Xún’s KR3fc045 Shìhú and KR3fc046 Shìlún. Dài Zhèn’s broader influence on the Yángzhōu circle — through his pupil 段玉裁 Duàn Yùcái’s introductions and Dài’s own brief Yángzhōu sojourn — was substantial.

Dating: NotBefore set at 1755 (allowing for the work to have been substantially complete during Dài Zhèn’s middle productive period); notAfter at 1773 (his Sìkù board appointment, after which his independent productive work largely ceased).

Translations and research

  • Jami, Catherine. 1992. “Une histoire chinoise du ‘nombre π’.” Archives Internationales d’Histoire des Sciences 38: 39–50. — Treats Dài Zhèn’s circle-cutting in the long history of Chinese π-computation.
  • Yü Ying-shih 余英時. 1976. Lùn Dài Zhèn yǔ Zhāng Xuéchéng 論戴震與章學誠. Hong Kong: Lóngmén shū-diàn. — Foundational treatment of Dài Zhèn’s intellectual project (philosophical and philological focus; treats the mathematical work as part of the broader evidential program).
  • Elman, Benjamin A. 1984. From Philosophy to Philology: Intellectual and Social Aspects of Change in Late Imperial China. Cambridge, Mass.: Council on East Asian Studies, Harvard University. — Provides the kǎo-zhèng context.
  • 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. 7.

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