Jígǔ suànjīng 緝古算經
Compilation of the Ancient Mathematical Classic by 王孝通 (Wáng Xiàotōng, fl. 618–626, 唐, zhuàn 撰; with the author’s own annotations)
About the work
Wáng Xiàotōng’s 1-juan independent mathematical treatise in 20 problems-with-solutions (shíjiǔ shù shíyī tí — actually 20 procedures across 11 distinct problem-types per the original presentation memorial), presented to the Tang Tàizōng emperor in the early Wǔdé period. The work is the principal Tang-period independent mathematical text and one of the most distinctive members of the Suànjīng shíshū — distinctive because, unlike the other members, it is not a Hàn-and-Six-Dynasties-period inheritance with later annotation but a fresh Tang-period composition reflecting Tang mathematical state-of-the-art.
The work’s substantive contributions:
(I) Cubic-equation problems (6 problems): six of the 20 procedures involve the formulation and solution of cubic equations from geometric considerations (typically right-triangle configurations where the area or volume relations yield a cubic in the unknown). Wáng Xiàotōng’s method is geometric-by-volume-decomposition rather than symbolic-algebraic, but the underlying mathematics is genuinely cubic. This is one of the earliest known systematic treatments of cubic equations in any tradition.
(II) Embankment, river-cutting, dyke, and granary computations (8 procedures): practical applications of Jiǔzhāng Shānggōng methods to specific Tang-period administrative problems involving labor-allocation for hydraulic engineering — the píngdì yìgōng shòu mào 平地役功受袤 problem of the Jiǔzhāng extended by Wáng Xiàotōng to handle non-rectangular configurations (sloped sides, varying widths, etc.).
(III) Square-and-cylindrical granary problems (5 procedures): paired procedures for square (fāngcāng) and cylindrical (yuánjiào) granaries.
(IV) Astronomical-calendrical problem (1 procedure): a procedure for computing the moon’s position at midnight on the new-moon night.
The work’s structural-and-pedagogical organization is notable: the 20 procedures are not arranged in order of increasing difficulty (the 提要 explicitly remarks on this) but by topical-and-administrative type. This makes the work somewhat difficult for the beginner (the 提要: “its order does not proceed by depth-and-shallowness, so the reader sometimes cannot quickly penetrate”) but well-organized for the working bureaucrat-mathematician.
The Tang Imperial Academy assigned a 3-year period for studying the Jígǔ — the longest study-period of any single member of the Suànjīng shíshū. The 提要 connects this long study-period to the work’s mathematical sophistication: “[knowing] this art’s refinement is not what can be exhausted in a morning-and-evening of its meaning”.
The Sìkù-recension descends from the Northern-Sòng Yuánfēng 7 (1084) imperial-secretariat collation by Zhào Yànruò 趙彦若, transmitted via Máo Yì 毛扆 (1640-1713)‘s Jígǔgé 汲古閣 transcription of a Zhāngqiū Lǐ family copy.
For other Suànjīng shíshū members, see KR3f0032, KR3f0033, KR3f0035, KR3f0036, KR3f0037, KR3f0038, KR3f0039. For the principal author, see 王孝通.
Tiyao
[Full 提要 in source file. Key points: The 提要 corrects the Tang Yìwén zhì and Sòng Chóngwén zǒngmù attribution of annotations to Lǐ Chúnfēng — the surviving recension’s annotations are in fact by Wáng Xiàotōng himself (“撰並注” zhuàn bìng zhù — composed and annotated [his own work]). The 提要 also corrects Wáng Yīnglín’s Yùhǎi claim that the work originally had 4 juàn with 3 lost — the 提要 demonstrates by comparing the yuán biǎo (presentation memorial)‘s “20 procedures” claim against the surviving 20 procedures that the work is in fact complete. The 提要 dated Qiánlóng 46 (1781), eleventh month.]
Translations and research
- Limited substantial secondary literature in European languages. Treated briefly in:
- 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.
Other points of interest
The 提要’s careful textual-critical work on the work’s authorship-and-completeness — correcting Tang Yìwén zhì / Sòng Chóngwén zǒngmù / Sòng Yùhǎi errors — exemplifies the high-Qīng kǎojù methodology applied to the Tang-period mathematical bibliographic record.
The work’s status as the only original Tang-period composition in the Suànjīng shíshū gives it special importance: it is direct evidence of Tang-period mathematical state-of-the-art, where the other members of the canon are Hàn-and-Six-Dynasties-period works with Tang-period annotation.