Wǔjīng suànshù 五經算術
The Five Classics’ Mathematical Methods by 甄鸞 (Zhēn Luán, Northern Zhōu, zhuàn 撰); imperially commissioned annotation by 李淳風 (Lǐ Chúnfēng, Tang, zhù 注)
About the work
Zhēn Luán’s 2-juan systematic mathematical commentary on the calculatable passages of the Five Classics — the foundational Confucian canon of Shūjīng, Shījīng, Yìjīng, Lùnyǔ (and the related Xiàojīng), Sānlǐ (the three Ritual Classics), Lǐjì, and Chūnqiū (with its Zuǒzhuàn commentary). For each passage in the classics that involves quantitative-and-computational reasoning, Zhēn Luán supplies the mathematical apparatus needed to verify or interpret the classical statement.
The work is structured topically by source-classic and by computational-problem-type:
Upper juàn: solar-and-lunar conjunction methods (tuī rìyuè hésù fǎ); 19-year intercalary cycle (qiú shíjiǔ nián qī rùn fǎ); the Shàngshū / Xiàojīng “myriad people” notation method (zhàomín zhù shù yuè cì fǎ); the Shī Fátán differing Máo and Zhèng commentaries; the Shī Fēngnián notation method; the Zhōuyì yarrow-stalk numbers; the Lùnyǔ “thousand-chariot state” calculation; the Zhōuguān chariot-cover method; the Yílǐ mourning-garment/cord measurements; the Lǐjì Wángzhì “ancient hundred-lǐ equals present 121-lǐ 60-bù 4-chǐ 2-cùn 2-fēn” method (and similar ZhèngXuán commentary verifications).
Lower juàn: Lǐjì Yuèlìng Huángzhōng pitch-pipe methods; Lǐjì Lǐyùn commentary methods; the Hànshū terminus-at-Nánshì method; Lǐjì Tóuhú (pitch-pot game) calculation; Chūnqiū Lǔ Xīgōng 5th-year zhèngyuè xīnhài shuò (1st-month new-moon date) method; the related “next-month new-moon” derivation; the Lǔ Xīgōng 5th-year winter-solstice computation; the related “next qì” derivation; Lǔ Wéngōng 1st-year (sìwèi / yǐwèi) intercalary-month placement reconstruction; Lǔ Wéngōng 6th-year intercalary-month problem; Lǔ Xiānggōng 27th-year intercalary problem; the Jiàngxiàn (Jìn) old man’s age-of-450-jiǎzǐ problem; Wéngōng 11th-year Jiàngxiàn old-man’s-birth-month problem; Lǔ Zhāogōng 19th-year intercalary-month placement; the related jírì derivation; Lǔ Zhāogōng 19th-year (wùyín) intercalary placement; Lǔ Zhāogōng 19th and 20th year monthly-new-moon and winter-solstice problems.
The work is therefore both a mathematical handbook and a Confucian-classical commentary; its purpose is to demonstrate the mathematical-arithmetic skills required for proper exegesis of the Five Classics, and conversely to demonstrate the canonical-textual grounding of the suànshù (computational arts) tradition. The Sìkù 提要 makes the explicit point that the work is “not only useful to the mathematical-arts family — it is also sufficient to clarify the Classics-and-Histories and to verify-and-correct doubtful meanings; for the kǎojù (evidential research) study it is especially meritorious”. This is one of the most explicit Sìkù editorial endorsements of the integration of mathematical-and-classical scholarship.
The Sìkù recension is recovered from the Yǒnglè dàdiǎn and reorganized into 2 juàn per the Tang Yìwén zhì’s recorded juan-count.
Tiyao
[Sub-classification: 子部, Tiānwén suànfǎ class 2, suànshū sub-category. Edition: WYG.]
Respectfully examined: Wǔjīng suànshù, 2 juàn. Zhēn Luán’s Wèishū and Hànshū commentaries record him as having served as Northern-Zhōu Shàng Yí Tóng Sānsī (Privy Equal-with-the-Three-Ducal-Offices), Hànzhōng Tàishǒu (commandery-administrator), having composed the Zhōu Tiānhénián lì and annotated the Jiǔzhāng, Wǔcáo, Sūnzǐ, and Zhōubì. [The histories] do not record his having this book.
The Suíshū Jīngjí zhì has Wǔjīng suànshù 1 juàn, [and] Wǔjīng suànshù lùyí 1 juàn — both not stating the [author’s] surname-and-name. The Táng Yìwén zhì then has Lǐ Chúnfēng’s annotated Wǔjīng suànshù 2 juàn — also not saying for whom the book was composed.
Now examining this book: it raises [from] the Shàngshū, the Xiàojīng, the Shī, the Yì, the Lùnyǔ, the Sānlǐ, the Chūnqiū’s computation-requiring [passages] which then clarifying [them] lists [them]; and the computation-pushing methods all add the words “[Zhēn] Luán’s note” at the head — then this book is what [Zhēn] Luán composed.
Furthermore, examining: [Lǐ] Chúnfēng during Zhēnguān (627-650) early-period [received] imperial decree, with the Suànxué bóshì Liáng Shù 梁述 [and] zhùjiào Wáng Zhēnrú 王真儒 et al. revising-and-fixing the Suànjīng, establishing [these texts] in the academy-office. The Táng Xuǎnjǔ zhì and Bǎiguān zhì both list the Wǔjīng suàn as one of the Suànjīng shíshū, with the Zhōubì together having a 1-year period [of study]; the testing of [examination] candidates each-raises-one-item as the question. This book’s annotation [openings] all have “subject [Lǐ] Chúnfēng et al. respectfully annotates” characters — then the Táng-period mathematics-section’s Wǔjīng suàn is just this book.
The book in the world has no transmitted recension; only it is dispersed-and-seen within the Yǒnglè dàdiǎn. Although [it has been] cut-up and lost-the-sequence, [it] still belongs to a complete book.
Based on Chúnfēng’s annotation to the Shàngshū tuīdìngrùn item, where he himself states the standard of his explanation, [we] then know [the work] begins from this. Furthermore, as in the Lùnyǔ qiānshèng zhī guó and the Zhōuguān gàigōngyǔqū — both use the kāifāng (square-root) method — xiángyúqiánérlüèyúhòu (detailed in the front and brief in the back), [we can] follow its [order-of-]meaning-and-example. According-to the various classics’ sequences [we] derive [the order]; [the work’s] old-order can be examined-and-known.
Respectfully [we] follow the Táng Yìwén zhì what is recorded as the count, ordering [it] into upper-and-lower 2 juàn. Within, [it] gathers-and-collects classics-and-histories, mostly Táng-and-prior-period old recensions: as it cites the Sīmǎ Biāo zhì preface discussing “the 12 pitch-pipes each rule-over one month, the corresponding-month each is its gōng (tonic)” — present-day HòuHàn zhì “tǒng (rule-over)” is wrongly-transcribed as “zhōng (terminate)”, “yuè (month)” wrongly-transcribed as “rì (day)”, “gémù zhī shēng (revolution-and-tree’s sound)” present-day zhì wrongly-transcribed as “cǎomù (grass-and-tree)”; “yángxiàshēngyīnyīnshàngshēngyáng (yang-bottom-gives-rise-to-yin yin-top-gives-rise-to-yang), beginning at Huángzhōng terminating at Zhōnglǚ” — present-day zhì lacks the “beginning at Huángzhōng” four characters; “the pitch-pipe makes cùn for the zhǔn (standard) makes chǐ*; the pitch-pipe makes* fēn for the zhǔn makes cùn*”; below the text continues zhǔncùn speaking of “not-filling [it] is the 1/10 obtained as fēn*” — present-day zhì lacks the “the pitch-pipe makes fēn for the zhǔn makes cùn*” two phrases. The Lǐjì yìshū citing the zhì is also wrongly-and-omittedly so. Furthermore, twice it cites: “upward-generating cannot exceed Huángzhōng ‘s muddy [pitch], downward-generating cannot reduce Huángzhōng ‘s clear [pitch]” — extending [this it] says: “then upward-generating cannot exceed 9 cùn, downward-generating cannot reduce 4 cùn 5 fēn”; with Cài Yōng’s Yuèlìng zhāngjù saying “Huángzhōng less-Tonic, the pipe-length 4 cùn 5 fēn” — agreeing! and sufficient to verify that [the] central-earth-pitch-pipe’s zhōng Huángzhōng zhī gōng is then the Huángzhōng clear-pitch [pipe], cannot be confused with the mid-winter month-pitch-pipe’s zhōng Huángzhōng as the longest muddy-pitch [pipe]. The Lǚshì chūnqiū first making Huángzhōng zhī gōng next making the 12-pipes — also [is] Huángzhōng having a clear-pitch’s verification.
Present-day zhì makes “upward-generating cannot exceed Huángzhōng ‘s qīngzhuó (clear-muddy), downward-generating cannot reach Huángzhōng ‘s number” — actually because the “qīng” character was wrongly-supplemented above; later people changed-and-cut-its-below — examined-against the pitch-pipe-method, [it] thereby becomes incommunicable. Indeed this book is not only what the mathematical-arts cannot abandon — [it is] also sufficient to clarify the Classics-and-Histories, examine-and-correct doubtful meanings; for the kǎojù study [it is] especially meritorious.
Respectfully collated, Qiánlóng 45, ninth month [October 1780].
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. 560–575 CE. Zhēn Luán was active under the Northern-Zhōu Wǔdì (560-578); his death is dated c. 575. The work was composed during this productive Northern-Zhōu period, before Lǐ Chúnfēng’s Tang-imperial annotation (c. 656).
The work’s significance:
(a) Mathematical-Confucian classical synthesis: the Wǔjīng suànshù is the principal pre-modern Chinese work that systematically applies mathematical analysis to the Confucian classical canon. By demonstrating that proper classical exegesis requires mathematical apparatus, the work positions mathematics as an essential component of Confucian scholarship rather than as a separate technical specialty. This integration of mathematics-and-classical-scholarship would shape Chinese mathematical practice through the late imperial period.
(b) Pitch-pipe and calendrical material as classical exegesis: the work’s substantial treatment of pitch-pipe (lǜlǚ) and calendrical (lì) computations demonstrates that these specialist domains are direct extensions of the canonical Confucian classical inheritance — the Lǐjì Yuèlìng and the Chūnqiū are the principal sources for both. This framing establishes the lǜlì tradition as part of mainstream Confucian scholarship, not a separate technical concern.
(c) The Sìkù editorial endorsement: the 提要’s strong statement that the work is “especially meritorious for kǎojù study” — citing the work’s preservation of pre-Tang readings of the HòuHànzhì and Lǐjì yìshū that have been corrupted in present-day transmission — places the Wǔjīng suànshù within the high-Qīng kǎojù (evidential research) project. This editorial framing is one of the more explicit endorsements of the integration of mathematical and philological scholarship in the Sìkù tíyào corpus.
For the parallel mathematical classics, see KR3f0032 Jiǔzhāng, KR3f0033 Sūnzǐ, KR3f0035 Hǎidǎo, KR3f0036 Wǔcáo. For the principal author and annotator, see 甄鸞 and 李淳風.
Translations and research
- Limited substantial secondary literature in European languages.
- Martzloff, Jean-Claude. A History of Chinese Mathematics, Berlin: Springer, 1997.
- Needham, Joseph (with Wang Ling), Science and Civilisation in China, vol. 3.
- Cullen, Christopher. Heavenly Numbers: Astronomy and Authority in Early Imperial China, Oxford: Oxford University Press, 2017.