Wǔcáo suànjīng 五曹算經
The Five Bureaus’ Mathematical Classic author unknown; annotated by 甄鸞 (Zhēn Luán, Northern Zhōu) and 李淳風 (Lǐ Chúnfēng, Tang)
About the work
A 5-juan elementary mathematical handbook organized topically around the practical computational concerns of the wǔcáo 五曹 — the “Five Bureaus” of imperial-administrative practice — one bureau per chapter:
- Juàn 1 Tiáncáo 田曹 (Field Bureau) — agricultural land mensuration
- Juàn 2 Bīngcáo 兵曹 (Military Bureau) — military supply-and-deployment computations
- Juàn 3 Jícáo 集曹 (Collection Bureau) — taxation collection-and-distribution
- Juàn 4 Cāngcáo 倉曹 (Granary Bureau) — grain-storage volume computations
- Juàn 5 Jīncáo 金曹 (Treasury Bureau) — currency-and-trade exchange computations
The work belongs to the same elementary-mathematical-handbook genre as the KR3f0033 Sūnzǐ suànjīng — providing topical problem-with-solution exercises at a more elementary level than the Jiǔzhāng. The Tang Suànxué guǎn curriculum paired the Wǔcáo with the Sūnzǐ as the elementary-introductory pair, with a one-year combined study period.
The work’s authorship is unknown. The Sui Jīngjí zhì lists a JiǔzhāngLiùcáo suànjīng 九章六曹算經 1 juàn (Nine-Chapters-and-Six-Bureaus, with 6 not 5 bureau-chapters); the Tang Yìwén zhì first lists a separately-titled Wǔcáo suànjīng in 5 juàn with annotations by Zhēn Luán (Northern Zhōu) and Hán Yán 韓延 (also a Northern Dynasties annotator) and the imperial annotation by Lǐ Chúnfēng. The Sìkù 提要 follows Zhēn Luán’s annotation as the terminus ante quem: since Zhēn Luán annotated the work, it must have existed before his death (c. 575 CE). The 5-bureau (vs. Sui-recorded 6-bureau) form may indicate a different recension from the JiǔzhāngLiùcáo one.
The Sìkù recension is recovered from the Yǒnglè dàdiǎn; the ZhēnLuán, HánYán, and LǐChúnfēng annotations have been lost in the transmission, but the main text is preserved completely.
Tiyao
Respectfully examined: The Suí Jīngjí zhì has JiǔzhāngLiùcáo suànjīng 1 juàn — but [it] has no Wǔcáo title; its 6-cáo chapter-titles are also not transmitted. The Táng Yìwén zhì first has Zhēn Luán’s Wǔcáo suànjīng 5 juàn, Hán Yán’s Wǔcáo suànjīng 5 juàn, [and] Lǐ Chúnfēng’s annotation [of the] Wǔcáo Sūnzǐ et al. suànjīng 20 juàn. The Mán-Hán-and-Zhēn two houses are both annotators of this book. As for the [original] composer, [we] do not know who.
Examining the Hànshū Méi Fú zhuàn: [Méi] Fú memorialized saying: “[I] heard [that] in Qí Huán-[gōng]‘s time, there was [the man] of the Jiǔjiǔ (Nine-Nine [tables])“. Yán Shīgǔ’s annotation: “the Jiǔjiǔ arithmetic-method [is] like the present-day Jiǔzhāng and Wǔcáo and so on”. For arithmetic-learning, although many, [it] does not exceed multiplication-and-division; multiplication-and-division [are] no more than [taken] from 1-to-9 in turn the jiǔzhī shù (nine-of-them numbers). Therefore [Méi Fú] raised the Jiǔjiǔ in his speech. And Shīgǔ accordingly with the [name of] the books at his time present — the Jiǔzhāng and Wǔcáo and so on — fits-them-into [Yán’s commentary]; not [meaning] that Méi Fú’s time had this book.
Zhū Yízūn’s Pùshūtíng jí has the Wǔcáo suànjīng bá (postface) saying: “tradition has it that the method comes from Sūn Wǔ. However, [Zhū] Yízūn merely says ‘tradition has it’, [without] cited evidence — indeed [this is] insufficient to depend [on]“. Examining the Tángshū Xuǎnjǔ zhì: it calls “[for the] Sūnzǐ [and] Wǔcáo [together,] one-year period [of study]“. Since [it] says “together-period”, then [the] Wǔcáo does not come from Sūnzǐ — it is clear.
Tentatively determining [it] by Zhēn Luán’s annotation: then this book is certainly before the Northern Qí period.
From the Yuán and Míng onwards, [there has been] long [no] printed recension; book-collecting houses’ transmitted-copies have transcription-errors, almost incommunicable. Now [it is] dispersed-and-seen within the Yǒnglè dàdiǎn: Zhēn Luán’s, Hán Yán’s, [and] Lǐ Chúnfēng’s annotations, although also dispersed-and-lost — but the classic-text [is] section-by-section completely-good. Now [we have] cross-referenced and collated, restoring the old appearance — accordingly [it has] become a juéwú jǐnyǒu (singular and rare) good edition.
Examining: the Xiàhóu Yáng suànjīng cites Tiáncáo and Cāngcáo twice, [and] Jīncáo once; but in this book all there is no [such] text. However, this book’s beginning-and-end clearly-divided, [its] threads-and-veins comprehensively-running, [it] also does not seem to have anything lost-or-missing. [We] suspect the Suí zhì’s JiǔzhāngLiùcáo — its headings perhaps [are] also similar to what [Xiàhóu] Yáng cited as Tiáncáo, Cāngcáo, Jīncáo, etc., names — these [are] then that book’s text, therefore [we] do not dare to depend [on this] in supplement-inserting them, [for fear of] confusing its truth.
Respectfully collated, Qiánlóng 47, fourth month [May 1782].
Abstract
Composition window: c. 400–570 CE. The 提要’s terminus ante quem of Zhēn Luán’s death (c. 575) is firm; the terminus post quem is less clear but Northern-Dynasties period is the conventional placement. The work belongs to the same broad genre and period as the KR3f0033 Sūnzǐ suànjīng.
The work’s significance is principally institutional-and-pedagogical rather than mathematically-innovative: as one of the Suànjīng shíshū, it shaped the Tang Imperial Academy curriculum at the elementary level, supplying topical problem-with-solution exercises organized around the practical concerns of imperial administration. Its 5-bureau organization (field, military, collection, granary, treasury) reflects the standard Tang-period fiscal-administrative division of imperial responsibilities.
For the parallel elementary-handbook, see KR3f0033 Sūnzǐ suànjīng. For other Suànjīng shíshū members, see KR3f0032 Jiǔzhāng, KR3f0035 Hǎidǎo, KR3f0037 Wǔjīng suànshù. For the principal annotators, see 甄鸞 and 李淳風.
Translations and research
- Limited substantial secondary literature. 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.