Héngzhāi suànxué 衡齋算學
Mathematical Studies from the Héng-zhāi Studio by 汪萊 (撰)
About the work
汪萊 Wāng Lái’s (1768–1813) collected mathematical studies in 8 juàn, transmitted under the studio-name Héngzhāi 衡齋. The work is the most theoretically-radical mathematical production of the JiāDào era and one of the principal documents of the Yángzhōu mathematical circle.
Abstract
Where the broader Yáng-zhōu-circle SòngYuán recovery project (張敦仁 Zhāng Dūnrén’s KR3fc041 Qiúyī suànshù, 焦循 Jiāo Xún’s KR3fc042–KR3fc048 Lǐtáng xuésuàn jì, 李銳 Lǐ Ruì’s equation-theory works) was directed primarily at recovering and re-expounding the indigenous mathematical methods, Wāng Lái’s Héngzhāi suànxué is directed at the substantive limits of those methods and at substantive new mathematics required to push beyond them.
The eight juàn cover (the precise composition varies by edition; the principal substantive content):
(1) Investigations of the Dàyǎn qiúyī shù (Chinese Remainder Theorem), with extensions to cases not treated by Qín Jiǔsháo.
(2) The theory of equations: the number-of-roots problem (how many real roots can a polynomial equation of given degree have, with given coefficients?) and the cases where the standard zēngchéng kāifāng method fails to produce a single root. Wāng Lái’s treatment, stated in indigenous terminology rather than as a general theorem, is in effect a partial Chinese-language anticipation of Descartes’s Rule of Signs and the Galois-style structural theory of polynomial roots.
(3) Right-triangle and circle-cutting problems, with novel solutions to several long-standing problem-types.
(4) Spherical-trigonometric studies, extending the Shìhú framework of Jiāo Xún.
(5–6) Algebraic-equation problems drawn from 李冶 Lǐ Yě’s Cèyuán hǎijìng and 朱世傑 Zhū Shìjié’s Sìyuán yùjiàn, with Wāng Lái’s independent treatments alongside the SòngYuán originals.
(7) The theory of the jiègēnfāng method, with systematic comparison to tiānyuányī.
(8) Miscellaneous mathematical investigations.
The work’s principal substantive contribution is the systematic engagement with the theory of polynomial equations beyond the procedural-computational kāifāng tradition. Wāng Lái’s recognition that the number of real roots of a polynomial equation depends substantively on the signs and magnitudes of the coefficients — and his attempt to formulate criteria for determining when standard procedures fail to find all roots — places him among the most theoretically-sophisticated mathematicians of his generation in any tradition.
His mathematical correspondence with 李銳 Lǐ Ruì in the years 1800–1813 — partly preserved in the Héngzhāi yíshū 衡齋遺書 and partly incorporated into the Suànxué — is the principal documentary record of high-level Jiā-Dào-era Chinese mathematical discussion.
Wāng Lái’s early death at age 45 (1813) deprived the Yángzhōu circle of its most theoretically-radical voice. His work was incorporated into subsequent SòngYuán recovery commentary (notably by 沈欽裴 and 羅士琳 in their Sìyuán yùjiàn commentaries) but the theoretical-substantive program he had begun was not fully developed within the indigenous tradition before the late-Qīng translation of European algebraic theory began to displace it.
Dating: the work is plausibly composed across Wāng Lái’s productive period — notBefore 1796 (allowing for early productive period after his early-1790s mathematical formation); notAfter his death year 1813.
Translations and research
- Hé Bǐngyù 何丙郁 (John Hoe). 1977. Les systèmes d’équations polynomes dans le Siyuan yujian (1303). Paris: Collège de France. — Treats Wāng Lái’s equation-theory in the context of the Jiā-Dào-era recovery.
- 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. 1999. Re-Kreation eines mathematischen Konzepts im chinesischen Diskurs. Stuttgart: Steiner.
- Lǐ Zhàohuá 李兆華. 2002. “Wāng Lái de shù-xué chéngjiù” 汪莱的数学成就. — Standard Chinese-language treatment.
- Tian Miao 田淼. 2003. “The Westernization of Chinese Mathematics: A Case Study of the Equation.” Historia Scientiarum 13.1.
Links
- Companion Yáng-zhōu-circle works: KR3fc041 (Zhāng Dūnrén), KR3fc042–KR3fc048 (Jiāo Xún), KR3fc060 (Lǐ Ruì)
- CBDB: https://cbdb.fas.harvard.edu/cbdbapi/person.php?id=79202