Cèyuán mìlǜ 測圓密率
Precise Ratios in the Measurement of the Circle by 徐有壬 (撰)
About the work
徐有壬 Xú Yǒurén’s (1800–1860) mathematical treatise in 3 juàn on the high-precision computation of trigonometric and cyclotomic ratios — the mìlǜ 密率 (“precise ratios”) of the circle that govern the relations between arc, chord, sagitta, and the inscribed-and-circumscribed polygons. The title echoes 祖沖之 Zǔ Chōngzhī’s famous Mìlǜ (the value π = 355/113), but the substance is squarely in the nineteenth-century cyclotomic-series tradition.
Abstract
The Cèyuán mìlǜ continues the indigenous Chinese cyclotomic-infinite-series tradition descending from 明安圖 Míng Āntú’s Gēyuán mìlǜ jiéfǎ 割圈密率捷法 through 董佑誠 Dǒng Yòuchéng (KR3fc068) and 項名達 Xiàng Míngdá (KR3fc069). Where Míng Āntú established the indigenous derivation of the trigonometric infinite series and Xiàng Míngdá theorised their unified origin (Xiàngshù yīyuán), Xú Yǒurén’s work is more focused on the computational application of the series to high-precision numerical results.
The 3-juàn treatise covers: (1) the basic chord-arc-sagitta infinite-series identities, presented in Xú Yǒurén’s own derivation; (2) the application of the series to high-precision tabular computation of π, of trigonometric functions, and of trigonometric integrals (in the sense of cumulative sums); (3) practical computational shortcuts and worked examples. The presentation is concise and computationally focused, in contrast to the more theoretical orientation of Xiàng Míngdá’s Xiàngshù yīyuán.
Xú Yǒurén’s other mathematical works (preserved in the Sìyuán wǔzhǒng 四元五種 collection but not separately in the present KR3fc series) extend the same approach to four-variable algebraic systems in the 朱世傑 Zhū Shìjié Sìyuán yùjiàn 四元玉鑑 tradition. He was a working member of the same mid-century mathematical circle as 戴煦 Dài Xù and 李善蘭 Lǐ Shànlán; the Cèyuán mìlǜ circulated in manuscript among these correspondents during the 1840s and 1850s.
Xú Yǒurén’s high official career (he rose to be Governor of Jiāngsū) and his mathematical work both ended with his death at the fall of Sūzhōu to the Tàipíng forces in 1860 — the same Tàipíng Tiānguó campaign in which 戴煦 Dài Xù was lost (Hángzhōu, also 1860) and 羅士琳 Luó Shìlín had been lost (Yángzhōu, 1853). The mid-century Tàipíng war effectively ended the indigenous mathematical lineage of the Chángzhōu / Sūzhōu / Hángzhōu schools.
Dating: Xú Yǒurén lived 1800–1860. notBefore 1830 (mature productive period); notAfter 1860 (death year).
Translations and research
- Tián Miǎo 田淼. 2003. Zhōng-guó shù-xué de xī-huà lì-chéng 中國數學的西化歷程.
- Horng, Wann-Sheng [洪萬生]. 1991. “Li Shanlan, the Impact of Western Mathematics in China during the Late 19th Century.” Ph.D. diss., City University of New York.
- Wú Wénjùn 吳文俊, ed. 1985. Zhōng-guó shù-xué shǐ dà-xì 中國數學史大系, vol. 8.
Links
- Cyclotomic-series tradition: KR3fc068, KR3fc069, KR3fc073
- CBDB (author): https://cbdb.fas.harvard.edu/cbdbapi/person.php?id=61310