聞分館君 Ask BunKanKun

四元玉鑑

4 min read

Text ID
KR3fc017
Date
1303–1303
Dynasty
元

Sìyuán yùjiàn 四元玉鑑

Jade Mirror of the Four Origins by 朱世傑 (撰)

About the work

朱世傑 Zhū Shìjié’s magnum opus and the principal monument of SòngYuán algebra, dated Dàdé 7 (1303). In 3 juàn containing 288 problems organized into 24 chapters, with the famous opening Gǔfǎ qīchéngfāng tú 古法七乘方圖 (Ancient-Method Eight-Powers Diagram) — the eight-row presentation of the binomial-coefficient triangle going back to Jiǎ Xiàn — and concluding with the Sìyuán shù 四元術 (four-element / four-unknown method) chapter. The most mathematically sophisticated pre-modern Chinese mathematical work and a foundational document of pre-modern algebra in any tradition.

Abstract

The Sìyuán yùjiàn extends 李冶 Lǐ Yě’s two-unknown tiānyuányī 立天元一 algebraic-equation method (KR3f0042 Cèyuán hǎijìng) to systems of up to four unknowns, designated tiān 天 (heaven), dì 地 (earth), rén 人 (man), and wù 物 (matter). The notational system arranges polynomial coefficients in a two-dimensional rod-numeral grid: the tiān row, the dì column, with rén and wù placed at the corner-positions, allowing the simultaneous representation of polynomial expressions in all four variables. Polynomial multiplication, addition, and elimination of variables are all performed by direct manipulation of the rod-grid; the most striking computational technique is the xiāofǎ 消法 (elimination procedure) for reducing a four-variable system to a sequence of single-variable equations, which is in effect a special-purpose ancestor of the resultant method that would not appear in European algebra until Bézout (1764).

Beyond the four-element method, the work systematizes:

(a) Finite-difference interpolation (zhāochā 招差): the chapter Rúxiàng zhāoshù 如象招數 develops the higher-order finite-difference formulas for interpolation, computing the n-th differences of tabulated functional values. The methodology in effect arrives at the Newton-forward-difference interpolation formula, some three centuries before Newton.

(b) Summation of figurate series (duǒjī 垛積): the chapter Duǒjī gē 垛積歌 gives closed-form expressions for sums of triangular numbers, square pyramids, and higher-order figurate sums up to and including Σ k(k+1)(k+2)(k+3) and Σ k²(k+1)². These are stated without proof but with full numerical verification, anticipating in special cases the Bernoulli-summation formulas.

(c) The systematic binomial-coefficient triangle: the opening Gǔfǎ qīchéngfāng tú gives the binomial coefficients up to the 8th power in triangular form. Zhū Shìjié explicitly attributes the triangle to “ancient method” (gǔfǎ), citing back to Jiǎ Xiàn 賈憲 of the Northern Sòng and via Yáng Huī’s KR3fc011 Xiángjiě jiǔzhāng suànfǎ.

Joseph Needham judged the Sìyuán yùjiàn “the most important mathematical work of the Chinese tradition” — both for its substantive content and for its position as the high-point of indigenous Chinese algebra before the late-Míng disruption.

Textual transmission and recovery: Zhū Shìjié’s sìyuán method was rapidly forgotten in the early Míng (the methodology was already opaque to 顧應祥 Gù Yìngxiáng in the mid-16th century); the work itself was effectively lost from active circulation by the late Míng. The Sìkù compilers were unable to recover a complete text. In the early Dàoguāng era a Korean recension was located and brought back to Yángzhōu, where 沈欽裴 Shěn Qīnpéi (see KR3fc019) and 羅士琳 Luó Shìlín (see KR3fc018) independently produced detailed xìcǎo (procedural exposition) commentaries that made the sìyuán method newly accessible to mid-Qīng working mathematicians. The collaborative re-publication appeared in the 1839 Yíjiātáng 宜稼堂 cóngshū, which established the modern Chinese text.

Dating: 1303 per Zhū Shìjié’s preface.

Translations and research

  • Hé Bǐngyù 何丙郁 (John Hoe). 1977. Les systèmes d’équations polynomes dans le Siyuan yujian (1303). Paris: Collège de France. — The standard Western-language monograph; full French translation with extensive mathematical commentary.
  • Hé Bǐngyù 何丙郁 (John Hoe). 2008. The Jade Mirror of the Four Unknowns by Zhu Shijie. New Zealand: Mingming Bookroom. — Subsequent partial English version by the same scholar.
  • Chemla, Karine. 1994. “Different Concepts of Equations in the Nine Chapters on Mathematical Procedures and in Sì-yuán yù-jiàn.” Historia Scientiarum 4-2: 113–137.
  • Martzloff, Jean-Claude. 1997. A History of Chinese Mathematics. Berlin: Springer.
  • Needham, Joseph (with Wang Ling). 1959. Science and Civilisation in China, vol. 3. Cambridge: Cambridge University Press.
  • Bréard, Andrea. 1999. Re-Kreation eines mathematischen Konzepts im chinesischen Diskurs. Stuttgart: Steiner. — Foundational discussion of Zhū Shìjié’s duǒ-jī (figurate-summation) methodology.

Graph View

Backlinks