Gēyuán liánbǐlì shù tújiě 割圈連比例術圖解
Geometric-Illustrative Exposition of the Cyclotomic Continued-Proportion Method by 董佑誠 (撰)
About the work
董佑誠 Dǒng Yòuchéng’s (1791–1823) mathematical treatise in 4 juàn presenting a systematic geometric explanation of the gēyuán liánbǐlì shù 割圈連比例術 — the “cyclotomic continued-proportion method” — the indigenous Chinese form of the infinite-series expansions for trigonometric functions (specifically the chord-arc relations) developed by 明安圖 Míng Āntú in the early eighteenth century. The work is a key document of the early-nineteenth-century Chángzhōu mathematical school.
Abstract
The mathematical content of the work derives from an extraordinary intellectual genealogy. The French Jesuit Pierre Jartoux (杜德美 Dù Démeǐ, 1668–1720), working at the Imperial Mathematics Bureau in Běijīng, communicated to his Chinese colleagues three trigonometric infinite-series identities (the series expansions for the chord in terms of the arc, the versed-sine in terms of the arc, and the arc in terms of the chord) — series that he had learned in Europe (apparently from James Gregory and John Machin’s work). Jartoux did not communicate the derivations of these series, only the formulae.
明安圖 Míng Āntú (1692?–1763?), a Manchu mathematician at the Imperial Mathematics Bureau, accepted Jartoux’s series as a challenge and worked for some thirty years to find indigenous-Chinese-style geometric derivations of them. His resulting treatise, the Gēyuán mìlǜ jiéfǎ 割圈密率捷法 (“Rapid Method for the Precise Ratios of Cyclotomy”), develops a systematic procedure using continued proportions of chords and segments of the circle to derive the series. Míng Āntú’s treatise was edited posthumously by his student Chén Jìxīn 陳際新 and the next-generation mathematician 張豸冠 Zhāng Zhìguān, and finally published in Dàoguāng 19 (1839) — substantially after Dǒng Yòuchéng’s death in 1823.
Dǒng Yòuchéng worked from the unpublished manuscript of the Gēyuán mìlǜ jiéfǎ, which had circulated in the Chángzhōu mathematical school via 李兆洛 Lǐ Zhàoluò and other channels. His 4-juàn Tújiě presents systematic geometric figures illustrating each step of Míng Āntú’s continued-proportion derivations, supplying the geometric intuition that the original treatise had assumed. The treatise is in effect a study-edition: making explicit the geometric reasoning behind Míng Āntú’s jiéfǎ (rapid procedure), and showing how the procedure produces the correct infinite-series expansions.
The work is one of the principal sources for the historical reconstruction of the Míng Āntú tradition — particularly important because Dǒng Yòuchéng worked from an earlier manuscript state of the Gēyuán mìlǜ jiéfǎ than the one ultimately printed in 1839. It is also a foundational document of the Chángzhōu mathematical school, paving the way for 項名達 Xiàng Míngdá’s KR3fc069 Xiàngshù yīyuán (a major theoretical extension of the cyclotomic-series tradition) and 戴煦 Dài Xù’s KR3fc073 tabular work.
Dating: Dǒng Yòuchéng lived 1791–1823. notBefore 1815 (mature productive period, age 24); notAfter 1823 (death year). The work was likely completed in the early Dàoguāng years.
Translations and research
- Jami, Catherine. 1990. Les méthodes rapides pour la trigonométrie et le rapport précis du cercle (1774). Paris: Mémoires de l’Institut des Hautes Études Chinoises. — Critical study and translation of Míng Āntú’s work.
- Jami, Catherine. 1988. “Western Influence and Chinese Tradition in an Eighteenth Century Chinese Mathematical Work.” Historia Mathematica 15: 311–331.
- Tián Miǎo 田淼. 2003. Zhōng-guó shù-xué de xī-huà lì-chéng 中國數學的西化歷程. Jǐ-nán: Shān-dōng jiāo-yù chū-bǎn-shè.
- Wú Wénjùn 吳文俊, ed. 1985. Zhōng-guó shù-xué shǐ dà-xì 中國數學史大系, vol. 7–8.