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詳解九章算法

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Text ID
KR3fc011
Date
1261–1261
Dynasty
宋

Xiángjiě jiǔzhāng suànfǎ 詳解九章算法

Detailed Exposition of the Nine Chapters Mathematical Methods by 楊輝 (撰)

About the work

楊輝 Yáng Huī’s 1261 detailed commentary on the KR3f0032 Jiǔzhāng suànshù (the foundational Chinese mathematical classic), originally in 12 juàn but transmitted in incomplete form. The catalog records the present recension as 2 juàn, reflecting the Sòng-fragment survival of only a portion of the original.

Abstract

Yáng Huī’s Xiángjiě was conceived as the systematic pedagogical reading of the Jiǔzhāng — supplying the detailed xìcǎo (working) for each of the original 246 problems, building on the 劉徽 Liú Huī and 李淳風 Lǐ Chúnfēng commentaries but extending their reasoning into a full step-by-step procedural exposition suited to a working mathematician. Yáng Huī’s preface (dated Xiánchún 1, 1261) characterizes the work as a xiángjiě 詳解 (detailed exposition) intended for the practical mathematical curriculum.

The work’s most famous content is the kāifāng zuòfǎ běnyuán tú 開方作法本源圖 (Diagram of the Source of the Method of Extracting Roots), appearing in the commentary on the Shǎoguǎng 少廣 chapter — the earliest known statement in any tradition of what is now called Pascal’s Triangle of binomial coefficients. Yáng Huī attributes the diagram to the now-lost Shìsuǒ suànshū 釋鎖算書 of Jiǎ Xiàn 賈憲 (Northern Sòng, fl. mid-11th century), pre-dating Pascal by some six centuries. The diagram shows the binomial coefficients C(n,k) arranged in triangular form up to the 6th power; Yáng Huī’s text supplies the procedural recipe for using the triangle to extract roots of any order (the zēngchéng kāifāng 增乘開方 method, also originating with Jiǎ Xiàn).

Beyond the Jiǎ Xiàn diagram, the Xiángjiě’s scholarly value lies in its preservation of pre-Yáng-Huī Sòng mathematical material. Yáng Huī’s source-citations record several lost Northern-Sòng mathematical works in addition to the Shìsuǒ suànshū, providing the principal documentation of mid-Sòng mathematical activity between Jiǎ Xiàn and the Southern-Sòng masters 秦九韶 Qín Jiǔsháo, 李冶 Lǐ Yě, and Yáng Huī himself.

Like the Yáng Huī suànfǎ (KR3fc010), the Xiángjiě was lost in China from the late Míng and recovered from a Korean recension by Yù Sōngnián in the 1842 Yíjiātáng cóngshū compilation. The KX edition descends from this line. The companion textual-critical apparatus is the KR3fc012 Xiángjiě jiǔzhāng suànfǎ zhájì of 宋景昌 Sòng Jǐngchāng, also a 1842 production.

Dating: 1261 per Yáng Huī’s preface.

Translations and research

  • Lam Lay-Yong. 1969. “On the Existing Fragments of Yang Hui’s Hsiang Chieh Suan Fa.” Archive for History of Exact Sciences 6: 82–88. — Foundational Western-language treatment of the survival and content of the Xiáng-jiě.
  • 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. — Central source on the Jiǎ Xiàn triangle and its mathematical implications.
  • Hé Bǐngyù 何丙郁 (John Hoe). 1977. Les systèmes d’équations polynomes dans le Siyuan yujian (1303). Paris: Collège de France. — Discusses the Xiáng-jiě as a Sòng-Yuán algebraic source.
  • Martzloff, Jean-Claude. 1997. A History of Chinese Mathematics. Berlin: Springer.
  • Bréard, Andrea. 2019. Nine Chapters on Mathematical Modernity: Essays on the Global Historical Entanglements of the Science of Numbers in China. Cham: Springer.

Other points of interest

The kāifāng zuòfǎ běnyuán tú triangle (binomial coefficients) makes the Xiángjiě a key document for global history-of-mathematics: it pushes back the documented appearance of the binomial-coefficient triangle from Pascal (1654) to Jiǎ Xiàn (mid-11th century), via the intermediate transmission through Yáng Huī (1261) and Zhū Shìjié (KR3fc017 1303). The eight-row triangle appearing in 朱世傑 Sìyuán yùjiàn is more elaborate than Yáng Huī’s six-row presentation, suggesting either direct transmission from Jiǎ Xiàn or progressive development within the SòngYuán tradition.

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