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求一術通解

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Text ID
KR3fc079
Date
1865–1876
Dynasty
清

Qiúyī shù tōngjiě 求一術通解

A General Exposition of the Qiú-yī Shù [Chinese Remainder Theorem] by 黃遵憲 (撰)

About the work

黃遵憲 Huáng Zūnxiàn’s (1848–1905) mathematical treatise in 4 juàn presenting a systematic general exposition of the qiúyī shù 求一術 — the indigenous Chinese procedure for solving the simultaneous-modular-congruence problem (the so-called Chinese Remainder Theorem), developed by 秦九韶 Qín Jiǔsháo in the Shùshū jiǔzhāng 數書九章 (KR3fc008) and extended by 駱騰鳳 Luò Téngfèng (KR3fc080 Yì yóu lù 藝游錄) and 時曰醇 Shí Yuēchún among others.

Abstract

The qiúyī shù (literally “seeking-one method”) is the indigenous Chinese label for the algorithm that solves the modular-congruence system: given moduli m₁, m₂, …, mₙ and remainders r₁, r₂, …, rₙ, find an integer x congruent to each rᵢ modulo each mᵢ. The problem is the classical Sūnzǐ 孫子 problem from the Sūnzǐ suànjīng 孫子算經 (KR3fa011): “Three threes leave two, five fives leave three, seven sevens leave two, what is the number?” The general procedure was developed by 秦九韶 Qín Jiǔsháo in Shùshū jiǔzhāng Chapter 1 (titled Dàyǎn shù 大衍術 “Great Extension Method”), using a sub-procedure called qiúyī shù — finding the multiplicative inverse modulo a given modulus.

The procedure was substantially forgotten in the Míng and was recovered as part of the QiánJiā evidential project on the indigenous Chinese mathematical tradition. The Qīng mathematical literature on the qiúyī shù includes works by 羅士琳 Luó Shìlín, 時曰醇 Shí Yuēchún, and 駱騰鳳 Luò Téngfèng (the latter’s Yì yóu lù in KR3fc080).

Huáng Zūnxiàn’s Qiúyī shù tōngjiě is a 4-juàn systematic and general (tōng 通) exposition of the procedure: the first juàn presents the theoretical foundation; the second and third present the procedure with worked examples covering the full range of cases (coprime moduli, non-coprime moduli, large-modulus cases); the fourth presents applications to calendrical and divination problems. The work is concise and computationally clear, distilling the substantial Qián-Jiā-to-Dào-guāng literature on the topic into a coherent treatment.

The work is dated by Huáng Zūnxiàn’s biography: it was composed during his jǔrén preparation period in the early-to-mid Tóngzhì years, before his entry to diplomatic service in 1877. The 4-juàn organisation and substantial bibliographic apparatus indicate it is a mature work rather than a youthful study; notBefore 1865 (early productive period), notAfter 1876 (year of jǔrén status, the conventional terminus for such pre-career mathematical writing). The work is sometimes given a publication date in the late 1870s under his developed diplomatic career, but the substantial content is the earlier jǔrén period.

Translations and research

  • Libbrecht, Ulrich. 1973. Chinese Mathematics in the Thirteenth Century: the Shu-shu chiu-chang of Ch’in Chiu-shao. Cambridge, Mass.: MIT Press. — Authoritative study of the qiú-yī shù tradition.
  • Martzloff, Jean-Claude. 1997 [2006]. A History of Chinese Mathematics. Berlin: Springer.
  • Wú Wénjùn 吳文俊, ed. 1985. Zhōng-guó shù-xué shǐ dà-xì 中國數學史大系, vol. 8.

Other points of interest

The work is unusual in Huáng Zūnxiàn’s larger oeuvre, which is dominated by poetry, diplomatic-historical writing on Japan and overseas societies, and political-reform tracts. The Qiúyī shù tōngjiě is his only substantial work in pure mathematics, and represents the jǔrén preparation period before his diplomatic career began. The combination of mathematical, diplomatic, and reform-political writing in a single career is characteristic of the late-Qīng jīngshì 經世 (“administering-the-world”) generation.

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