Zhāng Qiūjiàn suànjīng 張邱建算經
Zhāng Qiūjiàn’s Mathematical Classic by 張邱建 (Zhāng Qiūjiàn, 5th cent., 隋, zhuàn 撰); annotated by 甄鸞 (Zhēn Luán, Northern Zhōu, zhù 注); imperially commissioned re-annotation by 李淳風 (Lǐ Chúnfēng, Tang, fèngchì zhùshì 奉敕注釋); detailed procedures (xìcǎo 細草) by Liú Xiàosūn 劉孝孫 (Tang)
About the work
A 3-juan systematic mathematical classic in 100 problems-and-solutions, composed by Zhāng Qiūjiàn of Qīnghé in the late 5th or early 6th century. Structurally it follows the Jiǔzhāng problem-and-solution format but covers considerably more advanced material, with topics including:
- Juàn shàng (Upper): self-multiplication-and-division (chéngchú); right-triangle measurement (gōugǔ cèwàng) — problems 12-15 specifically on remote-sensing geometry; sum-and-difference relations of gōugǔ; redoubled-and-inverted gōugǔ configurations.
- Juàn zhōng (Middle): proportional distribution (chāfēn) and combined-with average-distribution (jūnshū); area-and-volume problems (fāngyuán mìjī); construction calculations (shānggōng); price-and-tax differences (guìjiàn chāfēn); arithmetic-progression problems; areal fāngtián fractional methods.
- Juàn xià (Lower): the famous Hundred Fowls problem (bǎijī wèntí); indeterminate-equation problems; deer-corral and granary three-volume problems with diagrams; húshǐ (arc-and-arrow) circular-segment problems.
The work’s most famous contribution is the Hundred Fowls problem: a chicken costs 5 qián, a hen costs 3 qián, three chicks cost 1 qián; with 100 qián one purchases 100 fowls. How many of each? This is a system of two equations in three unknowns with positive-integer constraints — i.e., a problem in indeterminate analysis. The solution-procedure given is the foundation of pre-modern Chinese indeterminate-equation analysis and the historical predecessor of what would become the European Diophantine analysis tradition (Diophantus c. 250 CE worked on similar problems, but his work was lost to medieval Europe and only rediscovered in the Renaissance; the Chinese tradition’s continuous engagement with such problems from Zhāng Qiūjiàn through Qín Jiǔsháo’s 13th-century elaboration is independent of any Hellenistic transmission).
The Sìkù recension descends from Máo Jìn’s 毛晉 (1599-1659) Jígǔgé 汲古閣 transcription of a Sòng manuscript fragment. The Sòng edition is the Northern Sòng imperial-secretariat collation by Zhào Yànruò 趙彦若, with the Zhēn Luán shù yuē 術曰 (procedure-says) annotations, the Liú Xiàosūn cǎo yuē 草曰 (procedure-elaboration-says) detailed working, and the Lǐ Chúnfēng chén … jǐn àn 臣…謹按 imperial-annotated comments preserved together. The Sìkù editors collated the Máo Jìn transcription against the Yǒnglè dàdiǎn fragments and supplied missing diagrams (problems 12-15 of the gōugǔ section, the missing arc-and-arrow problem in the middle juàn, the deer-corral and three-volume problems in the lower juàn).
The Sìkù 提要 explicitly commends the work for its connection to ancient Jiǔzhāng methodology: “[the work’s] questions-and-answers cross-collate and clarify; in total 100 conditions, condensed-and-ancient in style, rather like the Jiǔzhāng and unlike the recent methods. But its threads-and-organization [are] refined-and-dense, truly able to deeply investigate the ancients’ purport”. The Tang Imperial Academy thus made it a specialist text for advanced study.
For the broader Suànjīng shíshū context, see KR3f0032, KR3f0033, KR3f0035, KR3f0036, KR3f0037, KR3f0038. For the principal author, see 張邱建. For later Chinese indeterminate-analysis development, see Qín Jiǔsháo’s Shùshū jiǔzhāng (=KR3f0041 Shùxué jiǔzhāng).
Tiyao
[Full 提要 text in source file. Key points: dated Qiánlóng 46 (1781), fourth month. The 提要 places Zhāng Qiūjiàn in the early Sui period on internal-textual evidence (his preface refers to Xiàhóu Yáng’s and Sūnzǐ’s methods as predecessors). The 提要 details the multi-layered annotation: Zhēn Luán’s shù yuē (procedure-says); Liú Xiàosūn’s cǎo yuē (procedural-elaboration); Lǐ Chúnfēng’s imperial chén … jǐn àn (subject … respectfully notes) — only several dozen instances, not for every problem, only for those with doubtful interpretation. The Sìkù editors supplied missing diagrams for problems 12-15 (gōugǔ), retained the lacunas where the original text was incomplete, and corrected obvious copyist errors.]
Translations and research
- Lam Lay Yong and Ang Tian Se. Fleeting Footsteps, rev. ed., Singapore: World Scientific, 2004 (treats the Hundred Fowls problem genealogy).
- Martzloff, Jean-Claude. A History of Chinese Mathematics, Berlin: Springer, 1997.
- Libbrecht, Ulrich. Chinese Mathematics in the Thirteenth Century, Cambridge MA: MIT Press, 1973 (treats the indeterminate-analysis tradition Zhāng Qiūjiàn founded).
- Needham, Joseph (with Wang Ling), Science and Civilisation in China, vol. 3.