Liú Huī 劉徽
Active in the late CáoWèi 曹魏 / early Western-Jìn 西晉 period, third century CE. Birth and death years not securely recorded; dated by the Wèishū statement that he commenced his commentary on the Jiǔzhāng suànshù in Jǐngyuán 4 (263 CE) of the Wèi, but his commentary makes reference to the Jìn imperial-storehouse bronze hú (a Jìn-era unit-vessel) — so he was active before 263 (the commentary’s beginning) and continued after the WèiJìn transition (265 CE), making him at least a contemporary of the early Jìn. The Sìkù catalog conventionally classifies him as 晉; the Sìbù cóngkān (SBCK) edition of the Jiǔzhāng suànshù prefers 魏.
The single most important Chinese mathematician of the third century. His magnum opus is the systematic commentary (zhù 注) on the Jiǔzhāng suànshù (KR3f0032 Jiǔzhāng suànshù), which transformed that received compilation from a problem-collection into a coherent mathematical-theoretical work. Liu’s commentary supplied: (a) systematic geometric proofs (hémèi 合冪 / out-in compensating methodology) for the procedures stated without proof in the original; (b) the foundational gēyuán 割圓 (circle-cutting) method for computing π by inscribing successive regular polygons of doubling sides — Liú Huī computed π to 3.1416 via 192-gon, anticipating Zǔ Chōngzhī’s 5th-century refinement to 3.1415926; (c) the geometric account of right-triangle (gōugǔ) area-relations that anticipates the formal Pythagorean theorem; (d) extensive elaboration of the Fāngchéng (equation-systems) chapter, including what amounts to the systematic Gaussian-elimination algorithm in geometric form.
Liú Huī’s also-extant Hǎidǎo suànjīng 海島算經 (KR3f0035) — a 9-problem treatise on right-triangle remote-sensing techniques (computing the height of mountains from shadow-and-sighting ratios) — was originally an appendix-chapter of his Jiǔzhāng commentary (the Chóngchā 重差 chapter, mentioned in his preface to the Jiǔzhāng commentary), separately catalogued by the Tang as one of the Suànjīng shíshū 算經十書 (Ten Mathematical Classics).
His mathematical method is rigorous and self-consciously methodological. The preface to the Jiǔzhāng commentary articulates an explicit theory of fenced-and-derived mathematical demonstration (one root-method gives birth to many specific procedures by zhītiáo suī fēn ér tóng běngàn “the branches although divided yet share a single trunk”); the geometric proofs by visualizing area-and-volume rearrangements (chūrù hùxiāngbǔ 出入互相補) anticipate the methodology of Western proof-by-area-rearrangement that would become standard in European Renaissance mathematics. Liú Huī is conventionally regarded as the founding figure of theoretically-rigorous Chinese mathematics, and his work was the indispensable foundation for the subsequent Chinese mathematical tradition through the SòngYuán schools (Lǐ Yě, Zhū Shìjié, Yáng Huī, Qín Jiǔsháo) and into the MíngQīng synthesis.