Sūnzǐ suànjīng 孫子算經
Master Sūn’s Mathematical Classic attributed to “Master Sūn” (Sūnzǐ, identity uncertain — not Sūn Wǔ of the Art of War); originally annotated by 甄鸞 (Zhēn Luán, Northern Zhōu) and 李淳風 (Lǐ Chúnfēng, Tang) in the lost annotated recension
About the work
A 3-juan elementary mathematical classic of uncertain authorship and date, conventionally one of the Suànjīng shíshū 算經十書 (Ten Mathematical Classics) canonized by Lǐ Chúnfēng under the Tang Imperial Academy curriculum. The text consists of brief expositions of basic arithmetic operations followed by problems-with-solutions of progressively increasing difficulty, with topical content overlapping the Jiǔzhāng but at a more elementary level (proportional exchange, area-and-volume, taxation, transport, etc.). The work is famous for one specific contribution: the earliest known formulation of the Chinese Remainder Theorem (sūnzǐ dìnglǐ 孫子定理 in modern Chinese mathematical terminology), in problem 26 of the third juàn — the famous “things-of-which-the-number-is-not-known” (物不知其數) problem: an unknown number leaves remainders 2, 3, 2 when divided by 3, 5, 7; find the number. The solution-procedure given is the foundation of the modular-arithmetic method later systematized by Qín Jiǔsháo 秦九韶 in the 13th century.
The work’s authorship is unattributable. The traditional ascription to “Master Sūn” is not to be confused with the famous Sūn Wǔ of the Art of War (the early-Qīng bibliographer Zhū Yízūn 朱彝尊 examined this question and dismissed the conflation). Internal-textual evidence — the work mentions Chángān and Luòyáng as 900 lǐ apart (a Hàn-Wèi-period figure), and refers to a 29-chapter, 63-character-per-chapter Buddhist sūtra (a reference to the Hàn-period Buddhist transmission, post-Hàn Míngdì) — places its composition between the late Hàn and the Northern-and-Southern Dynasties (roughly 100–500 CE). The Sui Jīngjí zhì lists a 2-juàn recension; the Tang Yìwén zhì a 3-juàn annotated recension by Lǐ Chúnfēng on top of Zhēn Luán’s earlier annotation.
The Sìkù recension is reconstructed from the Yǒnglè dàdiǎn in 3 juàn, with the Yǒnglè-dàdiǎn-period editor Yáo Guǎngxiào having dropped the original Zhēn-Luán-and-Lǐ-Chúnfēng annotations (the Sìkù editors lament this loss).
Tiyao
[Sub-classification: 子部, Tiānwén suànfǎ class 2, suànshū sub-category. Edition: WYG.]
Respectfully examined: The Suí Jīngjí zhì has Sūnzǐ suànjīng 2 juàn, not stating its [author’s] name nor stating its time. The Táng Yìwén zhì says: Lǐ Chúnfēng annotated ZhēnLuán Sūnzǐ suànjīng 3 juàn — the zhènluán placed before sūnzǐ meaning that, similar to [Lǐ Chúnfēng]‘s annotation of the Zhōubì suànjīng, [Lǐ] further added discussion-and-debate to what [Zhēn] Luán had annotated.
The Suíshū’s discussion of shěndù (examining-degree) cites Sūnzǐ suànshù: “the silkworm’s birth-spitting silk makes a hū (tenth-of-a-second); shí hū wéi miǎo (10 hū make a miǎo ); shí miǎo wéi háo (10 miǎo make a háo ); shí háo wéi máo (10 háo make a máo ); shí máo wéi fēn (10 máo make a fēn )“. But the present book makes “10 hū make 1 sī (silk-strand); 10 sī make 1 háo”. Furthermore, [the Suíshū’s] discussion of jiāliàng (good-vessel-volumes) cites Sūnzǐ suànshù: “6 sù (grain) make a guī ; 10 guī make a chāo ; 10 chāo make a cuō ; 10 cuō make a sháo ; 10 sháo make a hé (unit-cup)“. But the present book makes “10 guī make 1 cuō; 10 cuō make 1 chāo; 10 chāo make 1 sháo”. Examining the Xiàhóu Yáng suànjīng: it cites Tiáncáo-and-Cāngcáo also like the present book. But within the Suíshū, what is cited often agrees with the histories-and-traditions. Indeed, ancient-book transmitted recensions are not unified; the collation-rectifying scholars each have their evidences — there is no harm in [letting] the divergences appear together.
The Tang’s xuǎnjǔ (selection-and-promotion) [examination] of the Suànxué used 10 books in total. Sūnzǐ and Wǔcáo together had a 1-year period of [study and] practice. Among later-period various arithmetic books, [the present book is] particularly close-to-the-ancient. Only it is not known what kind of person Sūnzǐ [was]. Zhū Yízūn 朱彝尊’s Jí (collected works) [of the Wǔcáo suànjīng postface] says: “tradition has it that the method comes from Sūn Wǔ. However, [Sūn] Wǔ separately has his own arithmetic-book; [for] those who examine the ancient, preserving the saying is acceptable”. Again [Zhū Yízūn] has the Sūnzǐ suànjīng postface saying: “The opening discusses the origin of measurement, agreeing with the military methods’ [statement of] earth-giving-rise-to-degree-degree-giving-rise-to-quantity-quantity-giving-rise-to-number; next discusses the methods of multiplication-and-division, setting up its numbers. The 13 chapters’ phrases — kuòdì fēnlì (opening-the-land dividing-the-profit), wěijī yuǎnshū (piling-up the distant-transport), guìmài bīngyì fēnshù (noble-selling military-service portion-numbers) — compared with the Jiǔzhāng ‘s fāngtián , sùmǐ , chāfēn , shānggōng , jūnshū , yíng bùzú headings, often correspond. And the essence is in ‘getting the calculation, much-much-calculation triumphs’. By this we know this compilation is not pseudepigraphy” — Yízūn’s intent indeed [was] to think [the work] truly comes from Sūn Wǔ.
Now examining within the book: the questions-set has “Chángān and Luòyáng are mutually distant 900 lǐ*”; also “Buddhist book 29 chapters, [each] chapter 63 characters” — these are post-Later-Hàn Míngdì people’s words. Sūn Wǔ was a man of the late Spring-and-Autumn period — how could he have such words? The old recension long lost; now from the Yǒnglè dàdiǎn what is recorded, [we have] gathered-and-edited, sequenced still as 3 juàn, headed by the original preface. As for the [annotations of] Zhēn-and-Lǐ two houses, [these are] no longer recoverable for examination — this is then Yáo Guǎngxiào and others’ fault of cutting-and-deleting overmuch.
Respectfully collated, Qiánlóng 43, seventh month [August 1778].
Chief Compilers: (subject) Jì Yún 紀昀, (subject) Lù Xíxióng 陸錫熊, (subject) Sūn Shìyì 孫士毅. Chief Collator: (subject) Lù Fèichí 陸費墀.
Abstract
Composition window: c. 100 CE – 500 CE. The 提要 fixes a terminus post quem of the late Hàn (post-Hàn-Míng-dì Buddhist reference) and an implicit terminus ante quem of the Suí (when the Jīngjí zhì lists the work). The Northern-and-Southern Dynasties period (4th–6th centuries) is the most likely composition window, with the work’s Sui Jīngjí zhì listing being the first secure documentary trace.
The work’s significance:
(a) The Sūnzǐ / Chinese Remainder Theorem: problem 26 of the third juàn — the wù bù zhī qí shù (things-of-which-the-number-is-not-known) problem — is the earliest known formulation of what would later be systematized as the Chinese Remainder Theorem. The problem asks for an unknown number leaving remainders 2, 3, 2 when divided by 3, 5, 7; the solution-procedure presented is essentially the modular-arithmetic algorithm that Qín Jiǔsháo 秦九韶 would generalize as Dàyǎn qiúyī shù 大衍求一術 in his 1247 Shùshū jiǔzhāng 數書九章. The Western mathematical tradition would not develop this procedure until Euler and Gauss in the 18th-19th centuries.
(b) Pedagogical canonization: as one of the Suànjīng shíshū, the Sūnzǐ suànjīng shaped East-Asian mathematical pedagogy from the Tang through the Sòng. Its more elementary character (relative to the Jiǔzhāng) made it suitable as an introductory text in the Tang Mathematics School curriculum — the “Sūn-zǐ-and-Wǔ-cáo together had a 1-year period” the 提要 records.
(c) Loss of the Zhēn-Luán-and-Lǐ-Chúnfēng annotations: the Sìkù-period editors specifically lament the Yáo Guǎngxiào / Yǒnglè-dàdiǎn-period editors’ decision to drop the original annotations. This is one of the few cases in the Sìkù tíyào where editorial criticism is directed at the Yǒnglè dàdiǎn compilers themselves.
For the related Jiǔzhāng, see KR3f0032. For other Suànjīng shíshū members, see KR3f0034, KR3f0035, KR3f0036, KR3f0037. For the related Tang annotators, see 李淳風 and 甄鸞.
Translations and research
- Lam Lay Yong and Ang Tian Se. Fleeting Footsteps: Tracing the Conception of Arithmetic and Algebra in Ancient China, rev. ed., Singapore: World Scientific, 2004 (treats the Sūn-zǐ suàn-jīng in detail).
- Martzloff, Jean-Claude. A History of Chinese Mathematics, Berlin: Springer, 1997.
- Libbrecht, Ulrich. Chinese Mathematics in the Thirteenth Century: The Shu-shu chiu-chang of Ch’in Chiu-shao, Cambridge MA: MIT Press, 1973 (treats the Chinese Remainder Theorem genealogy).
- Needham, Joseph (with Wang Ling), Science and Civilisation in China, vol. 3.
Other points of interest
The Sìkù 提要’s careful comparative-textual analysis — collating the present recension’s measurement-conventions against the Sui-shū’s earlier citations of Sūnzǐ suànshù and finding systematic differences — is one of the more refined exercises in textual genealogy in the Tiānwén suànfǎ 提要 corpus. The conclusion (different transmission-recensions diverged in the measurement-conventions, with the Sui-shū citation reflecting one variant and the present recension reflecting another) is methodologically sophisticated.