Yáng Huī 楊輝
Style name Qiānguāng 謙光. Native of Qiántáng 錢塘 (modern Hángzhōu). Birth and death years not securely recorded; conventionally placed fl. 1238–1298 on the evidence of his dated prefaces (his first dated work, Xiángjiě jiǔzhāng suànfǎ, is from Xiánchún 1, 1261; his last from c. 1275 by internal evidence). The conventional dates fl. 1238–1298 derive from inferred birthdate from his preface remarks and the date-range of his known works; CBDB lists multiple Yáng Huī entries without lifedates, none confidently identifiable with the mathematician.
A Southern-Sòng official-mathematician active in the lower-Yangzi region. His career is poorly documented; from internal evidence in his works he served as a minor local administrator and educator in Qiántáng / Línān (modern Hángzhōu), with documented appointments at the Tàizhōu jūnpàn 台州軍判 (Military Commissioner of Táizhōu) and at the Liútún 留屯 in Sūzhōu. His mathematical work was occasioned in part by his administrative duties.
One of the four great SòngYuán mathematicians, alongside 秦九韶 Qín Jiǔsháo, 李冶 Lǐ Yě, and 朱世傑 Zhū Shìjié. His mathematical corpus comprises five works, all written in the 1260s–70s and circulating as a collected series in the Yuán:
(1) Xiángjiě jiǔzhāng suànfǎ 詳解九章算法 (KR3fc011) — 12-juàn detailed commentary on the KR3f0032 Jiǔzhāng suànshù, completed 1261. (2) Rìyòng suànfǎ 日用算法 — daily-use calculation procedures (1262). (3) Yáng Huī suànfǎ 楊輝算法 (KR3fc010) — a 7-juàn compendium incorporating Chéngchú tōngbiàn běnmò 乘除通變本末 (1274), Tiánmǔ bǐlèi chéngchú jiéfǎ 田畝比類乘除捷法 (1275), and Xùgǔ zhāiqí suànfǎ 續古摘奇算法 (1275).
Yáng Huī’s principal mathematical contributions:
(a) The Jiǔzhāng commentary: his detailed exposition supplied much of the systematic reasoning that the 劉徽 Liú Huī commentary had stated only schematically; his methodology became the standard pedagogical-mathematical reading of the Jiǔzhāng in the Yuán and early Míng.
(b) The Jiǎxiàn / Yáng Huī triangle (binomial coefficients): in Xiángjiě jiǔzhāng 6.2, Yáng Huī cites a now-lost work Shìsuǒ suànshū 釋鎖算書 of Jiǎ Xiàn 賈憲 (Northern Sòng, fl. mid-11th century) for the kāifāng zuòfǎ 開方作法 triangle of binomial coefficients up to the sixth power. This is the earliest known statement of what is now called Pascal’s Triangle, predating Pascal by some six centuries.
(c) Magic squares: Yáng Huī’s Xùgǔ zhāiqí suànfǎ contains the first systematic Chinese treatment of magic squares (zònghéng tú 縱橫圖), including squares of orders 3 through 10. This is the foundational document of Chinese combinatorial mathematics.
(d) Pedagogical mathematics: Yáng Huī’s Chéngchú tōngbiàn běnmò is the first systematic Chinese exposition of mental-arithmetic techniques (the jiǎnjié 簡捷 shortcut methods), pre-figuring the abacus-based arithmetic of the Míng 程大位 Chéng Dàwèi tradition.