Shìhú 釋弧
Exposition of Arcs (Spherical Trigonometry) by 焦循 (撰)
About the work
焦循 Jiāo Xún’s monograph on spherical trigonometry in 4 juàn, part of his collected Lǐtáng xuésuàn jì (KR3fc042). The work systematizes the trigonometric methods needed for astronomical-spherical calculations, drawing on both the imported Jesuit-translated content and the indigenous gēyuán (circle-cutting) tradition.
Abstract
The Shìhú belongs to Jiāo Xún’s program of systematizing the imported Sino-European mathematical content for the Qián-Jiā-era working mathematician. Spherical trigonometry had been introduced to China through the Jesuit Lǐsuàn quánshū (KR3f0026) by 梅文鼎 Méi Wéndǐng (in his Húsānjiǎo jǔyú 弧三角舉要), the Lìxiàng kǎochéng (KR3f0018), and the Shùlǐ jīngyùn (1722); but no comprehensive Chinese-language exposition existed before Jiāo Xún.
The work’s four juàn:
(1) The foundational definitions: great-circle arc, sine and cosine of an arc, the zhèngshǐ 正矢 (versed sine), and the basic spherical-triangle vocabulary.
(2) The basic spherical-trigonometric identities — including the spherical law of sines, the spherical law of cosines (for sides and for angles), and the huíhú 回弧 (recursive-arc) relations — with proofs following the 安圖 Mínggatu power-series approach combined with the indigenous gēyuán circle-cutting derivations.
(3) Solution of spherical triangles: the systematic procedure for solving a spherical triangle given any three of its six elements (three sides, three angles), with worked examples drawn from astronomical applications (rising-and-setting of stars, latitude-and-longitude calculations).
(4) Applications: astronomical calculation problems — solar altitude, stellar position, parallax, eclipse-computation — drawn from the standard imperial astronomical procedures.
The work is the principal Qián-Jiā-era systematic Chinese-language treatment of spherical trigonometry, and remained the standard reference into the mid-19th century. 項名達 Xiàng Míngdá’s KR3fc070 and KR3fc071 works extend Jiāo Xún’s framework to more elaborate trigonometric problems.
For Jiāo Xún’s broader project see KR3fc042.
Dating: bracketed by Jiāo Xún’s productive period 1797–1820.
Translations and research
- Jami, Catherine. 2011. The Emperor’s New Mathematics. Oxford: Oxford University Press. — Treats the imperial-era spherical-trigonometric context.
- Roegel, Denis. 2010. “A reconstruction of Mei Wending’s tables of the sun, the moon, and the planets.” History of Mathematics Working Papers 26.
- Wú Wénjùn 吳文俊, ed. 1985. Zhōng-guó shù-xué shǐ dà-xì 中國數學史大系, vol. 7.
- Bréard, Andrea. 2019. Nine Chapters on Mathematical Modernity. Cham: Springer.