Shùxué jiǔzhāng 數學九章
(originally Shùshū jiǔzhāng 數書九章 — the Sìkù transmitted title appears to be corrupted) Mathematical Treatise in Nine Chapters by 秦九韶 (Qín Jiǔsháo, 1208–1261, 宋, zhuàn 撰), composed Chúnyòu 7 (1247)
About the work
Qín Jiǔsháo’s monumental 18-juan mathematical treatise in 9 topical chapters containing 81 problems-with-solutions, completed in Chúnyòu 7 (1247) — universally recognized as the supreme mathematical work of the SòngYuán period and one of the great mathematical works of any pre-modern tradition. The 9-chapter organization parallels the Jiǔzhāng suànshù (KR3f0032) in chapter-count but the topical content is entirely different, and the level of mathematical sophistication is dramatically higher.
The 9 chapters:
(I) Dàyǎn 大衍 (Great Expansion) — Chinese Remainder Theorem in systematic algorithmic form, the Dàyǎn qiúyī shù 大衍求一術 (Great-Expansion seeking-one method). This generalizes the KR3f0033 Sūnzǐ suànjīng problem-26’s specific procedure into a complete algorithm for solving any system of linear congruences with arbitrary moduli (including the difficult case where the moduli share common factors). The chapter is the foundational document of pre-modern Chinese number theory and one of the great algorithmic achievements of pre-modern mathematics. The 提要 specifically commends the chapter as the only place in the entire Suànjīng tradition that addresses the meaning (lìfǎ zhī yì 立法之意) underlying the procedure — i.e., gives a methodological-meta-statement about why the algorithm works.
(II) Tiānshí 天時 (Heavenly Times) — astronomical-calendrical computation: solar-and-lunar motion, planetary positions, gnomon shadows, the Wèishù (Star-Stem-Procedure) for planetary calculation. The 提要 criticizes some specific technical errors (particularly the conflation of Wèishù with the fāngchéng method) but acknowledges the chapter’s substantive astronomical content.
(III) Tiányù 田域 (Field Domains) — area calculations including triangular, trapezoidal, and curvilinear figures.
(IV) Cèwàng 測望 (Surveying) — height-and-distance measurement (extending the KR3f0035 Hǎidǎo tradition).
(V) Fùyì 賦役 (Taxation and Service) — administrative computation of taxes and labor.
(VI) Qiángǔ 錢榖 (Currency and Grain) — financial calculations.
(VII) Yíngjiàn 營建 (Civil Engineering) — earthwork volumes for hydraulic and construction projects.
(VIII) Jūnlǚ 軍旅 (Military Affairs) — formations and supply calculations.
(IX) Shìyì 市易 (Trade) — commercial calculations.
The work’s most consequential individual technical contribution is the systematic lì tiānyuányī shù 立天元一術 (“setting up the heavenly-element-one method”), the foundational Chinese algebra system that uses counting-rod arrangements to symbolically represent and manipulate polynomial equations. The 提要’s important observation: “the European new method [renamed it] jiègēnfāng and used it for the Nine Chapters and the Eight Lines [trigonometric tables] — its source actually opens from [Qín] Jiǔsháo*; this can also be said to be a contribution to mathematics*“. This is the classic Xīfǎ Zhōngyuán 西法中源 framing of European algebra: the European jiègēnfāng (introduced via Antoine Thomas, S.J., into the Kāngxī-period KR3f0018 Lìxiàng kǎochéng) is the rediscovery of Qín Jiǔsháo’s tiānyuányī.
The lì tiānyuányī method was used by Guō Shǒujìng 郭守敬 in the Shòushí lì’s arc-and-arrow (húshǐ) computations, and by Lǐ Yě 李冶 (KR3f0042 Cèyuán hǎijìng / KR3f0043 Yìgǔ yǎnduàn) for right-triangle and circle problems. Through this lineage, Qín Jiǔsháo’s algebra became the foundation of the SòngYuán mathematical tradition.
The 提要 acknowledges some specific weaknesses (the Shi-guà fāwēi part of the Dàyǎn chapter is criticized as misapplying Dàyǎn methodology to Yìjīng yarrow-stalk computation; the Tiānshí chapter’s astronomical computations use averaged rather than fixed seasonal-nodes and so are less accurate than post-Tycho European methods). But the verdict overall is highly favorable: the work has “both flaws-and-merits visible alongside, but the refined-and-correct passages are the majority”; Qín Jiǔsháo “during the Sòng’s late period alone rose-up and clarified the lost-learning”.
The Sìkù recension is recovered from the Yǒnglè dàdiǎn; the editors corrected obvious errors, supplemented missing material, reordered displaced sections, and added explanatory ànyǔ 案語 (editorial notes) under each problem.
For Qín Jiǔsháo’s biography, see 秦九韶. For the related SòngYuán mathematicians, see KR3f0042 Cèyuán hǎijìng by Lǐ Yě and KR3f0046 Yáng Huī suànfǎ by Yáng Huī. For the foundational source the Dàyǎn generalizes, see KR3f0033 Sūnzǐ suànjīng.
Tiyao
[Key points already summarized above. Full text in source file. Dated Qiánlóng 49 (1784), tenth month — relatively late in the Sìkù compilation.]
Translations and research
- Libbrecht, Ulrich. Chinese Mathematics in the Thirteenth Century: The Shu-shu chiu-chang of Ch’in Chiu-shao, Cambridge MA: MIT Press, 1973. The standard scholarly Western-language treatment.
- Mei Rongzhao 梅榮照, Míng-Qīng shù-xué-shǐ lùn-wén jí 明清數學史論文集, Nánjīng: Jiāngsū Jiào-yù Chūbǎnshè, 1990.
- Martzloff, Jean-Claude. A History of Chinese Mathematics, Berlin: Springer, 1997.
- Lam Lay Yong and Ang Tian Se. Fleeting Footsteps, rev. ed., Singapore: World Scientific, 2004.
- Needham, Joseph (with Wang Ling), Science and Civilisation in China, vol. 3 — extensive treatment.
- Cullen, Christopher. Heavenly Numbers: Astronomy and Authority in Early Imperial China, Oxford: Oxford University Press, 2017.
Other points of interest
The 提要’s characterization of the Sòng-period intellectual climate — “esteemed empty-discussion and despised practical-use; mathematics, although one of the Sage-Gate Six Arts, was likewise scorned and not spoken of… only Shěn Kuò exhausted his heart on this matter, and outside the Mèngxī bǐtán there was no completed book” — is a strikingly negative late-imperial Chinese assessment of Sòng intellectual culture’s relationship to mathematics. The framing isolates Qín Jiǔsháo (and the related SòngYuán mathematicians) as solitary figures in an otherwise mathematically-impoverished period.
The 提要’s Xīfǎ Zhōngyuán identification of Qín Jiǔsháo’s tiānyuányī with European jiègēnfāng algebra is the most influential statement of the SòngYuán recovered thesis in the Sìkù corpus. Through this framing, the 18th-century Chinese mathematical community could position the imported Western algebra as the rediscovery of native Chinese mathematics, rather than as a foreign import. Modern scholarship has tempered this claim (the European algebra developed independently and exceeded the SòngYuán in symbolic generality), but the underlying observation that the two traditions converged on equivalent symbolic-algebraic methods for polynomial equations is correct.