Wāng Lái 汪萊
Style name Xiàoyīng 孝嬰, sobriquet Héngzhāi 衡齋. Native of Shèxiàn 歙縣 (Huīzhōu 徽州 prefecture, Ānhuī). Born Qiánlóng 33 (1768); died Jiāqìng 18 (1813). CBDB id 79202 confirms the lifedates.
Jǔrén of Jiāqìng 6 (1801) — the same examination year as 焦循 Jiāo Xún — but never advanced to jìnshì; served briefly as instructor at the Shímén 石門 academy in Sūzhōu and as Hénán zhèngdìng sub-magistrate. Spent his life largely in private mathematical scholarship, in close contact with the Yángzhōu circle of Jiāo Xún, 張敦仁 Zhāng Dūnrén, and 李銳 Lǐ Ruì.
The most theoretically-radical mathematical scholar of the JiāDào era. Where the broader Yángzhōu circle (Zhāng Dūnrén, Jiāo Xún) worked primarily on the recovery of SòngYuán algebraic methods, Wāng Lái’s mathematical work focused on the substantive limits of the indigenous methods and the substantive new mathematics required to push beyond them.
His principal mathematical work is the Héngzhāi suànxué 衡齋算學 (KR3fc049) in 8 juàn — a collection of essays on various advanced mathematical topics. The work’s central substantive contribution is its treatment of the theory of equations: Wāng Lái was the first Chinese mathematician to systematically investigate the number-of-roots problem (how many real roots can a polynomial equation of given degree have, with given coefficients?) and to address the cases where the standard zēngchéng kāifāng method fails to produce a single root. His treatment, although 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.
Wāng Lái also engaged with the recovered tiānyuányī method (in correspondence with Lǐ Ruì) and contributed substantially to the qiúyī / Chinese-Remainder-Theorem literature. His mathematical correspondence with Lǐ Ruì in the years 1800–1813 — preserved in the Héngzhāi yíshū 衡齋遺書 — is the principal documentary record of high-level Jiā-Dào-era Chinese mathematical discussion.
His early death at age 45 deprived the Yángzhōu mathematical circle of its most theoretically-radical voice. Had he lived a normal span, the QiánJiā Chinese mathematical engagement with the substantive theory of equations might have produced results comparable to the contemporary European work; as it stands, Wāng Lái remains a remarkable foreshortened figure, his theoretical contributions not fully developed.