聞分館君 Ask BunKanKun

楊輝算法

5 min read

Text ID
KR3fc010
Date
1274–1275
Dynasty
宋

Yáng Huī suànfǎ 楊輝算法

The Yáng Huī Mathematical Methods by 楊輝 (編)

About the work

A composite mathematical compendium in 15 juàn compiled by 楊輝 Yáng Huī (fl. 1238–1298) in the 1270s, comprising three independently-titled works published in succession and subsequently transmitted as a single collection under the running title Yáng Huī suànfǎ. The three constituent works:

(1) Chéngchú tōngbiàn běnmò 乘除通變本末 (3 juàn, dated Xiánchún 10 = 1274) — the “Root-and-Branch of Multiplication-and-Division Methods”, an exposition of computational shortcuts, jiǎrú 假如 (mnemonic-pedagogical) problems, and the jiǔjiǔ 九九 multiplication table.

(2) Tiánmǔ bǐlèi chéngchú jiéfǎ 田畝比類乘除捷法 (2 juàn, dated Xiánchún 11 = 1275) — “Quick Methods for Field-Area Computations by Categorical Comparison”, an applied-mathematical treatment of land-measurement problems systematized by problem-type.

(3) Xùgǔ zhāiqí suànfǎ 續古摘奇算法 (2 juàn, also dated 1275) — “Selected-Curiosities Continuation of Ancient Calculation Methods”, the work’s most original mathematical content, containing Yáng Huī’s collection of curious problems (magic squares, indeterminate problems, recreational mathematics).

The collection has been transmitted partially in different recensions: the Korean Chosŏn recension preserves the full text (with Xùgǔ zhāiqí 1 juàn divided into 2 sub-juàn); the surviving Chinese editions descend from a Míng Yǒnglè dàdiǎn fragment supplemented from the Korean line. The KX edition presents the standard 15-juàn arrangement.

Abstract

The Yáng Huī suànfǎ is the principal pedagogical-mathematical work of the Southern Sòng, occupying a position parallel to but distinct from 秦九韶 Qín Jiǔsháo’s KR3f0041 Shùshū jiǔzhāng (1247) — where Qín’s work systematizes the high theoretical-algorithmic content of Sòng mathematics, Yáng’s work systematizes its pedagogical and recreational dimensions. The two works together with Yáng’s earlier KR3fc011 Xiángjiě jiǔzhāng suànfǎ (1261) constitute the principal mathematical output of the Southern-Sòng tradition.

Two of Yáng Huī’s substantive mathematical contributions appear in this collection. The first, in Xùgǔ zhāiqí suànfǎ, is the first systematic Chinese treatment of magic squares (zònghéng tú 縱橫圖), including squares of orders 3 through 10 (and a magic-circle of 33 numbers). Yáng Huī gives the construction principles for several of these by category — though he does not give a fully general algorithm for arbitrary order, his treatment of the luòshū 9-square, the jiǔgōng tú 4-square, and the systematic 5- and 6-squares establishes the basic framework of Chinese combinatorial mathematics. The second, scattered across the Chéngchú tōngbiàn běnmò, is the systematic jiǎnjié 簡捷 (shortcut-arithmetic) methodology, which would shape Míng abacus-arithmetic via 程大位 Chéng Dàwèi’s KR3fc027 Suànfǎ tǒngzōng.

The transmission history is unusually interesting. The work was lost in China from the late Míng onward; the Sìkù compilers were unable to recover a complete text. In 1842 the Yángzhōu scholar Yù Sōngnián 郁松年, working with 宋景昌 Sòng Jǐngchāng (the zhájì-author for the KR3fc012 Xiángjiě jiǔzhāng suànfǎ zhájì) and Sòng Jǐngchuān 宋景川, recovered a Korean Chosŏn-era recension brought back from Korea by the Japanese mathematician Seki Takakazu 関孝和 (in fact via the Edo-period mathematician Takebe Kenkō) and re-imported through the Lóngmén shūyuàn 龍門書院 channel. The 1842 Yíjiātáng 宜稼堂 cóngshū printing established the modern Chinese text — the same project that recovered the Shùshū jiǔzhāng and the KR3fc017 Sìyuán yùjiàn. The KX edition descends from this line.

Dating: each of the three constituent works carries its own preface-date (1274 and 1275 respectively); notBefore and notAfter are bracketed accordingly.

Translations and research

  • Lam Lay-Yong. 1977. A Critical Study of the Yang Hui Suan Fa: A Thirteenth-Century Chinese Mathematical Treatise. Singapore: Singapore University Press. — The standard English-language monograph, with full translation of substantial portions of the Yáng Huī suàn-fǎ and extended commentary; indispensable.
  • Lam Lay-Yong. 1980. “The Chinese Connection between the Pascal Triangle and the Solution of Numerical Equations of any Degree.” Historia Mathematica 7: 407–424.
  • Andrews, W. S. 1917. Magic Squares and Cubes, 2nd edn. Chicago: Open Court. — Includes the early Western awareness of Yáng Huī’s magic squares.
  • Cammann, Schuyler. 1960. “The Evolution of Magic Squares in China.” Journal of the American Oriental Society 80: 116–124. — Foundational Western-language treatment.
  • Wú Wénjùn 吳文俊, ed. 1985. Zhōng-guó shù-xué shǐ dà-xì 中國數學史大系, vol. 5. Beijing: Běi-jīng shī-fàn dà-xué chū-bǎn-shè. — Standard modern Chinese-language reference.

Other points of interest

The Xùgǔ zhāiqí suànfǎ preserves the first Chinese-language statement of what is now called the Hundred Fowls generalization — a class of indeterminate-equation problems building on the KR3f0039 Zhāng Qiūjiàn suànjīng’s original. Yáng Huī’s treatment is procedural rather than theoretical; the full algorithmic generalization appears only later, in the Dàyǎn qiúyī shù of 秦九韶 Qín Jiǔsháo.

Graph View

Backlinks