Shìtuǒ 釋橢
Exposition of Ellipses by 焦循 (撰)
About the work
焦循 Jiāo Xún’s monograph on the ellipse in 2 juàn, part of the collected Lǐtáng xuésuàn jì (KR3fc042). One of the earliest substantive Chinese-language treatments of conic sections.
Abstract
The ellipse (tuǒyuán 橢圓 or tuǒ 橢) had become relevant to Chinese astronomical practice through the transmission of Keplerian planetary theory in fragmentary form in the eighteenth century — particularly through the writings of 蔣友仁 Michel Benoist (in his Kūnyú quántú 坤輿全圖 and elsewhere) and through the imperial astronomical compilations of the post-1742 Lìxiàng kǎochéng hòubiān (the supplement to KR3f0018 that incorporated revised European astronomical procedures). Jiāo Xún’s Shìtuǒ is the principal Qián-Jiā-era systematic Chinese-language treatment.
The work’s two juàn:
(1) The basic geometry of the ellipse: the two-focus definition, the chángzhóu 長軸 (major axis) and duǎnzhóu 短軸 (minor axis), the eccentricity, the area formula, and the basic geometric properties (the tangent-and-normal at a point, the chord-and-diameter relations). Jiāo Xún provides Euclidean-style demonstrations following the methodology of 徐光啟 Xú Guāngqǐ’s KR3fc026 Gōugǔ yì.
(2) Astronomical applications: Kepler’s first law (planetary orbits as ellipses with the sun at one focus), Kepler’s second law (the area-rate law, which Jiāo Xún derives geometrically), and the practical procedures for computing planetary positions from elliptical orbits. The treatment is non-trivial — Jiāo Xún provides the xīnchā fǎ 心差法 (eccentric-correction procedure) for converting between mean motion and true motion on the ellipse, which is the central computational problem of elliptical-orbit astronomy.
The work is one of the few Qián-Jiā-era Chinese mathematical works to engage substantively with the post-Keplerian European astronomy. It anticipates by some decades the more systematic Keplerian treatments that would appear in the late-Qīng mathematical literature (notably 李善蘭 Lǐ Shànlán’s mid-19th-century work and the KR3fc068 / KR3fc073 / KR3fc076 works of the mid-19th-century mathematical generation).
The work also represents Jiāo Xún’s most theoretical-mathematical production — engaging with conic-section geometry rather than the more familiar arithmetic and algebra of his other volumes. Together with the Shìhú and Shìlún, it constitutes Jiāo Xún’s systematic Chinese-language version of the imported Sino-European mathematical-astronomical content.
For Jiāo Xún’s broader project see KR3fc042.
Dating: bracketed by Jiāo Xún’s productive period 1797–1820.
Translations and research
- Hashimoto Keizō 橋本敬造. 1988. Hsü Kuang-ch’i and Astronomical Reform. Osaka: Kansai University Press.
- Jami, Catherine. 2011. The Emperor’s New Mathematics. Oxford: Oxford University Press.
- Sivin, Nathan. 2009. Granting the Seasons. New York: Springer.
- 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.