Suànshū shū 算數書
Book of Mathematical Procedures by anonymous (闕名; text preserved as bamboo-slip manuscript)
About the work
A Han-dynasty mathematical handbook preserved on approximately 190 bamboo strips discovered in 1983–1984 at Zhangjiashan 張家山 (tomb no. 247), Jiangling County 江陵縣, Hubei Province. The tomb was sealed not later than 186 BCE, placing the text among the earliest extant Chinese mathematical manuscripts. The 算數書 is not a transmitted text with WYG or other conventional editions; it belongs to the category of excavated documents (chutu wenxian 出土文獻) alongside the Beida Qin Suànshū shū and the Qinghua Suànbiǎo 算表. The Kanripo text file is a modern transcription of the bamboo slips.
The work covers a wide range of practical mathematical topics organised in short, self-contained sections, each bearing a section-head (mùtí 目題). The section structure visible in the Kanripo transcription includes: fractions and multiplication (相乘 xiāng chéng, 分乘 fēn chéng, 增減分 zēngjiǎn fēn, 約分 yuēfēn, 合分 héfēn, 徑分 jìngfēn); commercial problems (出金 chū jīn, 共買材 gòng mǎi cái, 狐出關 hú chū guān, 狐皮 hú pí, 息錢 xī qián, 漆錢 qī qián, 繒幅 zēng fú, 賈鹽 gǔ yán); grain conversion and granary management (舂粟 chōng sù, 粟求米 sù qiú mǐ, 米求粟 mǐ qiú sù, 粺毇 bài suì, 耗 hào, 程禾 chéng hé); transport and labour (負炭 fù tàn, 負米 fù mǐ, 傳馬 chuán mǎ, 婦織 fù zhī, 女織 nǚ zhī, 籚 lú); and geometry (旋粟 xuán sù, 囷蓋 qūn gài, 圜亭 yuán tíng, 井材 jǐng cái, 啟廣 qǐ guǎng, 啟縱 qǐ zòng, 少廣 shǎo guǎng, 大廣 dà guǎng, 方田 fāng tián, 里田 lǐ tián). Many sections present one or more worked examples followed by a formula or procedure (shù 術).
The transcription preserves editorial annotations “楊” and “王” after certain sections, indicating verification marks by multiple collators (the names, likely of the scribes or checkers, appear at the end of blocks they verified): “楊已讎” and “王已讎” mean “Yang (or Wang) has collated [this portion].” This multi-scribe feature is characteristic of Qin-Hàn bureaucratic document production.
Abstract
Discovery and physical description. The 算數書 was unearthed in the winter of 1983–1984 from Zhangjiashan tomb no. 247 in Jiangling County, Hubei. The tomb also contained the legal compendium 奏讞書 and other texts; it was sealed in the early Han (lǚ hòu 呂后 era, shortly after 187 BCE), establishing 186 BCE as a terminus ante quem for all contents. The complete manuscript totals approximately 7,000 characters on 190 strips. The standard annotated transcription with photographs was edited by Péng Hào 彭浩 and published as Zhāngjiāshān Hànjian Suànshū shū zhùshì 張家山漢簡算數書注釋 (Kēxué, 2001; critically reviewed by Jì Zhìgāng 紀志剛 in Zìrán kēxué yánjiū 2004.1: 91–96).
A separate Suànshū shū of Qin date (ca. 214 BCE) was found among the Peking University Qin bamboo slips acquired in 2010; it is a related but independent text. The Kanripo KR3f0065 transcription corresponds to the Zhangjiashan version.
Dating and composition. The content — practical administrative arithmetic relating to tax grain conversion rates, commercial pricing, land measurement, and construction volumes — reflects the concerns of Qin-Hàn county-level officials. Individual problems and procedures likely circulated before the Qin unification (221 BCE); the final compilation in the tomb copy dates to late Warring States or early Hàn (ca. 300–186 BCE). The composition bracket notBefore −300 / notAfter −186 reflects this range; the 186 BCE terminus is the only hard date.
Mathematical content and significance. The 算數書 predates by over two centuries the earliest transmitted mathematical classic, the Jiǔzhāng suànshù 九章算術 (Nine Chapters, KR3f0032), yet shares many of the same problem types and procedural vocabulary, including the surplus-and-deficit (盈不足 yíng-bùzú) method, proportional distribution (shuāifēn 衰分), and the “join fractions” (héfēn 合分) algorithm. This overlap has been central to scholarly debate on how the Jiǔzhāng tradition developed from earlier administrative mathematics. The 算數書 is also of exceptional value for economic history: it contains the earliest datable rates for grain conversion yields, land-tax procedures, interest calculation, and commodity prices in the early Han period (Wilkinson §38.17.1; Yè Yùyīng 葉玉英, Zhōngguó shèhuì jīngjìshǐ yánjiū 2005.1: 38–45).
The fractions treatment is notably sophisticated: the text presents multiple algorithms for reducing, adding, and multiplying fractions, including a Euclidean-algorithm-like procedure for greatest common divisor reduction (約分術) that does not appear explicitly in the transmitted Jiǔzhāng until much later.
Relation to other KR3f texts. As an excavated Han text without a traditional commentarial transmission, the 算數書 does not belong to the Suànjīng shíshū 算經十書 (Ten Mathematical Classics) assembled in the Tang and preserved through the Sìkù. The most closely related transmitted texts in KR3f are KR3f0032 (Jiǔzhāng suànshù 九章算術) and KR3f0033 (Sūnzǐ suànjīng 孫子算經), all of which share the shù 術 (procedure) and jīn yǒu 今有 (problem-statement) formulaic patterns.
Translations and research
- Cullen, Christopher. 2004. The suàn shù shū “Writings on reckoning”. English translation with explanatory commentary and Chinese text. Needham Research Institute Working Paper 1 (posted online). — First full English translation.
- ⸻. 2007. The suàn shù shū, “Writings on reckoning”: Rewriting the history of early Chinese mathematics in the light of an excavated manuscript. Historia Mathematica 34.1: 1–44.
- Dauben, Joseph W. 2008. Suan shu shu: A book of numbers and computations: English translation with commentary. Archive for History of Exact Sciences 62: 91–178.
- Péng Hào 彭浩, ed. 2001. Zhāngjiāshān Hànjian Suànshū shū zhùshì 張家山漢簡算數書注釋. Kēxué. — Standard Chinese annotated edition with photographs.
- Anicotte, Rémi. 2017. Fractions in the Suàn shù shū (China, beginning of the 2nd century BCE). JCL 45.1: 20–67.
- Yè Yùyīng 葉玉英. 2005. Lùn Zhāngjiāshān Hànjian Suànshū shū de jīngjì shǐliào jiàzhí 論張家山漢簡算數書的經濟史料價值. Zhōngguó shèhuì jīngjìshǐ yánjiū 1: 38–45.
- Zou Dahai 鄒大海. 2017. An elementary study of the unearthed mathematics book, Suanshu shu. Chinese Annals of History of Science and Technology 1.1: 24–58.
- Chemla, Karine. 2022. How do excavated manuscripts and transmitted canons and commentaries shed light on each other? An outlook from mathematics. EC 45: 269–301.
- Wilkinson, Endymion. Chinese History: A New Manual, 6th edn., 2022, §38.17.1 (primary sources for Chinese mathematics) and §38.17.3 (readings).
Other points of interest
The collation marks “楊已讎” (Yáng yǐ chóu) and “王已讎” (Wáng yǐ chóu) — “Yang/Wang has completed the collation” — that appear at section boundaries in the Kanripo transcription are among the rare surviving examples of scribal quality-control annotations in a pre-imperial administrative document. They imply that the bamboo-book was prepared by at least two named scribes (surnamed 楊 and 王) who each checked the other’s work, a practice documented in Qin and early Han administrative law for official records.
The 里田 lǐtián section at the end of the Kanripo text (calculating fields measured in lǐ) uses the formula 廣一里縱一里為田三頃七十五畝 (one lǐ wide, one lǐ long = 3 qǐng 75 mǔ), which presupposes the 240-bù mǔ standard established by the Shang Yang reforms in Qin (350 BCE) and extended to the whole empire after 221 BCE. This is consistent with the Zhangjiashan text’s early Han date and Wilkinson’s observation (§36.1) that the 240-bù mǔ was universal from 221 BCE to 1911.