Jiǔzhāng suànshù 九章算術
Nine Chapters on the Mathematical Art attributed (by tradition) to 張蒼 (Zhāng Cāng, early Hàn) and 耿壽昌 (Gěng Shòuchāng, mid-Hàn) for compilation; with the foundational commentary by 劉徽 (Liú Huī, 魏-Jìn, zhù 注, Jǐngyuán 4 = 263 CE); imperial annotation by 李淳風 (Lǐ Chúnfēng, 唐, zhùshì 注釋); and phonetic-and-meaning gloss by 李籍 (Lǐ Jí, 唐, yīnyì 音義, 1 appended juàn)
About the work
The foundational mathematical classic of the Chinese tradition, in 9 juàn containing 246 problems with their solutions, organized into nine topical chapters covering the principal branches of pre-modern Chinese applied mathematics:
- (I) Fāngtián 方田 (Square Fields, 38 problems) — area calculations for rectangular, triangular, trapezoidal, circular, and other plane figures; including the foundational treatment of fractions and π (taken as 3 in the original but corrected by Liú Huī to 3.14 and by Zǔ Chōngzhī to 3.1415926).
- (II) Sùmǐ 粟米 (Grain-Rice, 46 problems) — proportional exchange of commodities (the Rule of Three in elementary arithmetic).
- (III) Cuīfēn 衰分 (Distribution by Proportion, 20 problems) — proportional distribution problems.
- (IV) Shǎoguǎng 少廣 (Less-Width, 24 problems) — areas-and-roots: extraction of square-and-cube roots, particularly through area-decomposition methods.
- (V) Shānggōng 商功 (Construction Computations, 28 problems) — volume calculations for prisms, pyramids, cones, frustra, and spheres.
- (VI) Jūnshū 均輸 (Equitable Transport, 28 problems) — taxation, transport, and labor-distribution calculations involving complex weighted averages.
- (VII) Yíng bùzú 盈不足 (Surplus-and-Deficit, 20 problems) — the Rule of False Position method for solving linear equations.
- (VIII) Fāngchéng 方程 (Rectangular Arrays, 18 problems) — systematic solution of systems of linear equations by what is in effect Gaussian elimination performed on counting-rod arrangements; the chapter introduces negative numbers as a standard mathematical concept (the first known systematic treatment in any tradition).
- (IX) Gōugǔ 勾股 (Right-Triangle, 24 problems) — the Pythagorean theorem and its applications, including remote-sensing techniques and the famous “broken-bamboo” and “leaning-ladder” problems.
Plus a 10th Yīnyì 音義 (Phonetic-and-Meaning) appended juàn by 李籍 (Lǐ Jí, Tang).
The work is the principal foundational classic of pre-modern Chinese mathematics, comparable in importance to Euclid’s Elements in the Greek tradition. Its received recension reflects multiple layers of compilation-and-annotation:
- Original compilation (early Hàn): traditionally attributed to Zhāng Cāng 張蒼 and Gěng Shòuchāng 耿壽昌, both early-Hàn officials with mathematical interests. Internal evidence (the work’s references to Chángān Shànglín 長安上林, the imperial park established under Wǔdì in the 130s BCE) shows that the received text cannot be earlier than the mid-Western-Hàn (c. 100 BCE).
- The Liú Huī commentary (魏-Jìn, 263+ CE): Liú Huī of the WèiJìn period composed the foundational commentary, supplying systematic geometric proofs for the procedures stated baldly in the original. His preface (preserved in the SBCK recension) traces the work back to Zhōu Gōng’s Six Arts curriculum and acknowledges the early-Hàn redaction by Zhāng Cāng and Gěng Shòuchāng. Liú Huī also supplied the Chóngchā 重差 (Double-Differences) supplementary chapter, which would later be detached as the KR3f0035 Hǎidǎo suànjīng under the Tang.
- The Lǐ Chúnfēng imperial annotation (唐, 656 CE): Lǐ Chúnfēng under Táng Gāozōng’s commission compiled-and-annotated the Suànjīng shíshū 算經十書 (Ten Mathematical Classics), of which the Jiǔzhāng suànshù is the principal text. His annotations clarify the Liú Huī commentary at points where it had become obscure.
- The Lǐ Jí phonetic-and-meaning gloss (唐, late 9th century): Lǐ Jí supplied the Yīnyì appended juàn, providing phonetic readings and meaning-glosses for difficult terms.
- Recovery from Yǒnglè dàdiǎn (Sìkù period, 18th century): the work had nearly disappeared from circulation by the late Míng; the Sìkù editors recovered most of it from the Yǒnglè dàdiǎn. Some chapters had to be supplemented from other sources.
The work’s substantive significance for Chinese mathematics is comprehensive. It establishes the basic problem-types and computational procedures that would shape Chinese mathematical practice for two millennia. Its Fāngchéng chapter’s introduction of negative numbers — and its systematic Gaussian-elimination procedure for systems of linear equations — predates the comparable European developments by over a millennium. Its Gōugǔ chapter’s treatment of right-triangle problems is one of the earliest systematic geometric treatises in any tradition.
Tiyao
[Sub-classification: 子部, Tiānwén suànfǎ class 2, suànshū sub-category. Edition: WYG.]
Respectfully examined: Jiǔzhāng suànshù — for it is the bequeathed method of the Bǎoshì (Tutor-of-Skills) of the Zhōulǐ; not knowing by whom it was transmitted. The Yǒnglè dàdiǎn cites the Gǔjīn shìtōng 古今事通 saying: Wáng Xiàotōng 王孝通 said, “The Zhōu Gōng established the lǐ and there was the name ‘nine chapters’ — its principle is hidden-and-subtle, its form secret-and-condensed. Zhāng Cāng cut-and-supplemented its remnants-and-gaps; verifying its categories, [they were] rather different from the ancient art” — and so on.
Now examining within the book: there is the name of Chángān Shànglín (the Imperial Hunting Park at the Hàn capital). The Shànglín park was [established] in Wǔdì’s time; Cāng [lived] at the early Hàn — by what reason would [he] presciently have included [it]? We know that those who composed this book [were] in the middle-or-later [years] of the Western Hàn.
The old recension has commentary, signed by Liú Huī. Examining the Jìnshū: it says that in the Wèi Jǐngyuán 4 [263 CE], Liú Huī commentated the Nine Chapters. However, the commentary refers to “the Jìn imperial-storehouse bronze hú” — then [Liú] Huī after entering the Jìn further added supplements.
Furthermore there is a zhùshì (annotation), signed by Lǐ Chúnfēng. Examining the Tángshū: it says [Lǐ] Chúnfēng et al. by imperial decree annotated the Jiǔzhāng suànshù as the head of the Suànjīng shíshū. The Imperial-Academy-Bureau established 30 algebra-students [to] practice the Jiǔzhāng and the Hǎidǎo suànjīng, with a 3-year period limit — indeed [Lǐ Chúnfēng’s annotation] was composed at this time.
From the Northern Sòng onward, [the work’s] technique [had been] rarely transmitted… Of mathematical books none is more ancient than the Jiǔshù (Nine Numbers); of Nine Numbers none is more ancient than this book.
[The 提要 here continues at length, mostly preserved through the Sìkù transmission. The full text traces the work’s citations in earlier bibliographic catalogs and assesses its place in the Chinese mathematical tradition.]
Abstract
Composition window: c. 100 BCE – 263 CE. The base-text was compiled in the mid-to-late Western Hàn (the Chángān Shànglín reference fixes a terminus post quem of approximately 110 BCE); the Liú Huī commentary added to the received recension was composed beginning in 263 CE and continued into the Western-Jìn period after 265.
The work’s significance:
(a) Foundational classic of Chinese mathematics: alongside the KR3f0001 Zhōubì suànjīng, the Jiǔzhāng suànshù is one of the two foundational pre-modern Chinese mathematical works. Through the Jiǔzhāng, the Chinese tradition’s distinctive applied-mathematical heritage — proportional exchange, area-and-volume calculation, root extraction, system-of-equations solution, right-triangle problems — entered the canonical record. The 246 problems would shape the form of Chinese mathematical exposition (problem-with-solution) for two millennia.
(b) The Liú Huī commentary: Liú Huī’s contribution transformed the received compilation from a procedural collection into a theoretically-rigorous work. His foundational accomplishments include (i) the gēyuán 割圓 (circle-cutting) method for π, computing π = 3.1416 via 192-gon (extended to 3072-gon for higher precision in the surviving fragments); (ii) systematic geometric proofs (chūrù hùxiāngbǔ 出入互相補, “outward-and-inward mutual-compensation”) for the area-and-volume formulas of the Fāngtián and Shānggōng chapters; (iii) the explicit articulation of negative-number arithmetic in the Fāngchéng chapter; (iv) the Chóngchā 重差 supplementary chapter on remote-sensing techniques (later detached as the KR3f0035 Hǎidǎo suànjīng).
(c) The Lǐ Chúnfēng (Tang) editorial-and-pedagogical canon: Lǐ Chúnfēng’s 656 imperial annotation made the Jiǔzhāng the first text of the Suànjīng shíshū (Ten Mathematical Classics), the official curriculum of the Tang Imperial Academy’s Suànxué guǎn 算學館 (Mathematics School). Through this institutional canonization, the Jiǔzhāng shaped East-Asian mathematical pedagogy for centuries (the same canonization led to the work’s transmission to Korea and Japan as part of the standard mathematical curriculum).
(d) The negative-number breakthrough: the Fāngchéng chapter’s systematic treatment of negative numbers — represented in the original as red-and-black counting-rod color-conventions — is the first known systematic mathematical use of negative numbers in any culture. The European mathematical tradition would not systematically embrace negative numbers until the 16th-17th centuries.
(e) The Sìkù recovery: the work’s near-disappearance by the late Míng and its Yǒnglè dàdiǎn-based recovery in the 18th century is one of the major Sìkù-period textual restorations. The Sìkù-recension’s collation against pre-Sòng manuscript fragments and Tang-period epigraphic citations established the canonical text used by modern scholarship.
For the mathematical-historical context, see KR3f0001 Zhōubì suànjīng (the parallel astronomical-mathematical classic). For the related Liú Huī work, see KR3f0035 Hǎidǎo suànjīng. For other Tang-period Suànjīng shíshū members, see KR3f0033 Sūnzǐ suànjīng, KR3f0034 Shùshù jìyí, KR3f0036 Wǔcáo suànjīng, KR3f0037 Wǔjīng suànshù. For the principal commentators, see 劉徽 and 李淳風.
Translations and research
- Chemla, Karine, and Guo Shuchun 郭書春. Les Neuf Chapitres: Le classique mathématique de la Chine ancienne et ses commentaires, Paris: Dunod, 2004. The standard scholarly French edition with exhaustive philological apparatus and modern French translation; the indispensable reference work for any serious study.
- Shen, Kangshen, John N. Crossley, and Anthony W.-C. Lun. The Nine Chapters on the Mathematical Art: Companion and Commentary, Oxford / Beijing: Oxford University Press / Science Press, 1999. The standard English translation.
- Guo Shuchun 郭書春. Jiǔ-zhāng suàn-shù xīn-jiào 九章算術新校, Hé-féi: Zhōng-guó Kē-jì Dà-xué Chū-bǎn-shè, 2014. The principal modern Chinese critical edition.
- Cullen, Christopher. Astronomy and Mathematics in Ancient China: The Zhou Bi Suan Jing, Cambridge: Cambridge University Press, 1996 (parallel context).
- Li Yan 李儼 and Du Shiran 杜石然. Chinese Mathematics: A Concise History, trans. John N. Crossley and Anthony W.-C. Lun, Oxford: Clarendon, 1987.
- Martzloff, Jean-Claude. A History of Chinese Mathematics, Berlin: Springer, 1997 (revised English edition; standard general reference).
- Needham, Joseph (with Wang Ling), Science and Civilisation in China, vol. 3, Cambridge: Cambridge University Press, 1959 (foundational treatment).
Other points of interest
The 提要’s identification of internal-textual evidence (the Chángān Shànglín reference) for dating the work’s redaction post-Wǔ-dì is one of the more refined pieces of late-imperial Chinese textual-historical analysis in the Sìkù tíyào corpus. The Sìkù editors’ willingness to use anachronistic-reference methodology to establish dating of received texts — quite rare in late-imperial Chinese textual scholarship — anticipates modern philological methods.
Liú Huī’s preface (preserved in the SBCK recension) — with its tracing of mathematical knowledge back through Zhōu Gōng’s Six Arts to the legendary Bāoxī shì (Fúxī)‘s eight-trigrams — articulates a distinctive Chinese mathematical-historical theory in which mathematics is a central component of the canonical Confucian curriculum, alongside ritual, music, archery, charioteering, and writing. This framing places mathematics within the liùyì (Six Arts) tradition rather than as a separate technical specialty, an intellectual positioning that would shape Chinese mathematical practice through the late imperial period.
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