Lìsuàn quánshū 歷算全書
Complete Book of Calendrical Computation by 梅文鼎 (Méi Wéndǐng, 1633–1721, 清, zhuàn 撰); posthumous compilation by Wèi Lìtóng 魏荔彤 and Yáng Zuōméi 楊作枚, with additions
About the work
The canonical anthology of Méi Wéndǐng’s mathematical-and-astronomical writings, in 60 juàn containing 29 separate sub-treatises (shíèrshíjiǔ zhǒng 二十九種). Méi Wéndǐng composed the works over five decades (c. 1670–1721, his entire mature career); the anthology was assembled posthumously by Wèi Lìtóng (a wealthy patron-of-scholarship from Bǎimén 柏門) who obtained Méi’s papers and entrusted them to Yáng Zuōméi of Wúxī for editing-and-supplementation. Yáng added his own commentary and supplied missing material before publication c. 1726. The Sìkù-preserved recension is reorganized by the Sìkù editors (the original WèiYáng arrangement was held by the editors to be “sequentially confused, not having grasped the principal threads”), with calendrical-astronomical works placed first and mathematical works following.
The reorganized contents (per the 提要) are:
Calendrical-astronomical (前 / first half):
- Lìxué yíwèn 歷學疑問 (3 juàn) — the foundational philosophical-historical work; presented to Kāngxī in 1702 with imperial approbation; opens the anthology
- Lìxué yíwèn bǔ 歷學疑問補 — supplementary calendrical theory
- Lìxué dáwèn 歷學答問 — letters of calendrical exchange with the gōngqīngdàifū (high officials)
- Hú sānjiǎo jǔyào 弧三角舉要 — spherical trigonometry, with sphere-instrument diagrams
- Huánzhōng shǔchǐ 環中黍尺 — the spherical-trigonometric “angle-substituting-for-computation” method
- Suìzhōu dìdù hékǎo 嵗周地度合考 — apsidal-and-yearly mensuration; comparison of Western years-months-and-degree-units
- Pínglì dìng sānchā shuō 平立定三差説 — explanation of the seven-regulators’ surplus-and-shrinkage
- Dōngzhì kǎo 冬至考 — winter-solstice examination using Tǒngtiān, Dàmíng, Shòushí methods
- Zhūfāng rìguǐ 諸方日軌 — solar-altitude tables for latitudes 20°–42°
- Wǔxīng jìyào 五星紀要 — planetary motion overview
- Huǒxīng běnfǎ 火星本法 — Mars-specific (Tycho’s reform was originally for Mars)
- Qīzhèng xìcǎo 七政細草 — practical computation procedures
- Kuírì hòuxīng jìyào 揆日候星紀要 — Zhílì, Jiāngnán, Hénán, Shǎnxī shadow-and-stellar-position tables
- Èrmíng bǔzhù 二銘補註 — annotations on the Yǎngyí míng and Jiǎnyí míng (Guō Shǒujìng-period instrument-inscriptions)
- Lìxué piánzhī 歷學駢枝 — annotated Dàtǒng calendrical-method
- Jiāohuì guǎnjiàn 交會管見 — eclipse-position reform from north-south-east-west to up-down-left-right
- Jiāoshí méngqiú 交食蒙求 — eclipse-computation procedures
Mathematical (後 / second half):
- Gǔ suàn yǎnlüè 古算衍略 — extension of ancient mathematics
- Chóu suàn 籌算, Bǐ suàn 茟算, Dù suàn shìlì 度算釋例 — three computational-instrument primers (rod-counting, Napier’s-bones [imported via Jesuits], proportional compass)
- Fāngchéng lùn 方程論 — discussion of equation systems
- Gōugǔ chǎnwēi 勾股闡微 — right-triangle theorems
- Sānjiǎofǎ jǔyào 三角法舉要 — plane-trigonometry essentials
- Jiě gēyuán zhī gēn 解割圜之根 — circle-section root-explanation
- Fāngyuán mìjī 方圓冪積 — square-and-circle areas
- Jǐhé bǔbiān 幾何補編 — Euclid supplement
- Shǎoguǎng shíyí 少廣拾遺 — Shǎoguǎng completion (one of the Nine Chapters’s technical sections)
- Qiàndǔ cèliàng 塹堵測量 — solid-geometry measurement (with the qiàndǔ prismatic-figure)
The 提要’s enthusiastic verdict (the most fulsome any Sìkù tíyào* gives a private scholar in the Tiānwén suànfǎ category): “from Guō Shǒujìng onward, [his] equal has rarely been seen”; “self-receiving the Sage Son-of-Heaven’s special-distinguishing recognition was indeed not by chance”.
Tiyao
[Sub-classification: 子部, Tiānwén suànfǎ class 1, tuībù sub-category. Edition: WYG.]
Respectfully [examined]: Lìsuàn quánshū, 60 juàn, by Méi Wéndǐng of Our Dynasty. Wéndǐng’s zì was Dìngjiǔ 定九, [a man of] Xuānchéng. Dǔzhì shìgǔ (firmly-purposeful, fond-of-the-ancient), [he was] especially refined in the calendrical-and-arithmetic learning. In Kāngxī 41 [1702], Grand Secretary Lǐ Guāngdì once presented his Lìxué yíwèn to the throne. It happened that the Sage Ancestor Benevolent Imperial Ancestor [Kāngxī] was on a southern progress at Dézhōu, and summoned [Méi] for an audience. The imperially-written Jīxué cānwēi 積學參微 (“accumulated learning, threading the subtle”) in four characters was bestowed on him; on account of [his] old age [he was] sent home. Subsequently, the imperial decree to compile the Lǜlǚ lìsuàn shū descended to the Jiāngnán Governor-General; [the imperial command] summoned his grandson [Méi] Juéchéng [Méi Juéchéng 梅㲄成] to enter palace service. After the Lǜlǚ zhèngyì book was completed, [Méi Wéndǐng] further received the imperial command via post-station to verify-and-collate. Years later, at over 90 years old, [he] died at home. The Sage [Kāngxī] specifically commanded the [Imperial] Textile Bondservant Cáo Fǔ to manage the funeral arrangements — to this day [this] is transmitted as the supreme honor of examining the ancient.
The various calendrical-arithmetic books he composed: Lǐ Guāngdì once printed seven titles. The remainder, mostly compositions of his late years, [are either] already revised-into-volumes or roughly-having-the-draft. Wèi Lìtóng obtained their copies and entrusted [them] to Yáng Zuōméi of Wúxī for collation-and-correction. [Yáng] Zuōméi accordingly appended his own theories; [together] for [Méi] supplementing what was incomplete, and printed [it] for circulation. In total 29 titles, named the Lìsuàn quánshū. However, the sequential-arrangement was confused, not grasping the principal threads.
Respectfully heavy-edited [we have re-organized]: those who are calendrical-establishers placed in front, those who are calculation-discussers placed behind.
[The 提要 lists each of the 29 titles with brief content notes — see the section above for the reorganized list.]
In general, [the work treats] the explication of arithmetic methods, either expanding what the Nine Chapters had not completed, or noting the surface-figures of the present method, or discussing the variations of Chinese-and-Western body-shapes, or explaining the proportions of arc-and-sine and right-triangle and eight-lines [trigonometry]. Indeed [with this], the technique of calendrical-arithmetic at this point is greatly complete.
Our State, having cultivated-clear the calendrical-numbers, probed-the-deep and sought-the-hidden, gathered the great-completion of a thousand-ancient. [Méi] Wéndǐng, as a cǎoyě shūshēng (rural village book-student), was able to cover-thoroughly the investigation, deeply-knowing the source-flow. What he discussed and composed are all sufficient to penetrate the purport of Chinese-and-Western and bend [it] to the middle of present-and-ancient. From Guō Shǒujìng onward, [his] equal has rarely been seen. His receiving the Sage Son-of-Heaven’s special-distinguishing recognition was indeed not by chance.
Respectfully collated, Qiánlóng 46, tenth month [November 1781].
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.
Lǐ Guāngdì’s Gōngjì 恭記 (Reverently-Recorded, 1704)
In rénwǔ [1702], 10th month, [I] accompanied the imperial chariot on the southern progress, halting at Dézhōu. There was an imperial command to fetch the books that had been engraved [under my supervision]. [I] reported that, because of haste, [I] had not brought them; furthermore, the books mostly were classics-and-examination-time-essays, response-school-tutoring’s needs — not sufficient to dust the imperial gaze. There was Mr. Méi Wéndǐng of Xuānchéng, Lìxué yíwèn in 3 juàn, that [I] had revised-and-engraved. [I] respectfully submitted [it], seeking the Sage Instruction.
[Imperial] decree said: “The Imperial Self has paid attention to calendrical-arithmetic for many years; in this matter the Self can decide its right-or-wrong. Will leave the book for inspection. After two days, [the bearer will] receive a face-to-face [audience]“.
The Above said: “The book submitted yesterday is very fine-hearted; furthermore the discussion is also impartial. This man has used [his] effort deeply. The Self [will] take [it] back to the palace and inspect-and-read in detail”.
I therefore requested that the August Above personally add imperial brush refutation-and-revision-and-fixing — so that the rural-and-village scholar would have something to take-and-cut. I myself also obtained to share-in-hearing one or two — fortunate beyond endurance.
The Above assented to [this]. Crossing into the next year’s spring, the imperial chariot again [made the] southern progress, and accordingly, at the imperial residence, sent back the original book.
Face-to-face spoken: “The Self has already detail-by-detail read [it]. Within, the circles, dots, smearings, and the slip-attached comments — all are by the Imperial brush”. Subjectively I again asked where this book had defects. The Above said: “No defects, but the computational methods are not yet complete” — for the Méi book originally was not finished.
The Sage Verdict accordingly reached this. Privately I hold: from antiquity, scholars of Way-and-enterprise who, having received the imperial decree, had something composed — not to mention them. As for the private-house storage-records, [these are] mostly buried-in-dust and bùfù (jar-coverings). As for the calendrical-imagery and celestial-officer’s depths, [these are] especially what the Confucians of the world call zhuānmén juéxué (specialist abstruse learning). For [those of] the fond-of-the-strange and indulging-in-the-rare type, [they] often cannot finish a chapter and leave-off — how could [they] reach to trouble the Yǐyè (second-watch)‘s viewing, line-by-line discussion and character-by-character debate, mutually responding like teacher-and-disciple? Master Méi’s encounter may be called thousand-loads in a moment (a once-in-a-millennium occurrence).
Now the Imperial Hand-writing circulates throughout the realm; only there has not yet been a book cut-by-the-Sage-hand (personally edited by the imperial hand) stored among men. How is this only the Luòxià is-or-is-not is firmly fixed? Even Zǐ Yún’s bequeathed compilations — what is called encountering an age’s ruler, surpassing all the masters — also need not [wait] for Huán Tán’s repeated sigh!
Having returned the book to Master Méi, [I] write its time-month, accordingly raise [it] [for the record], so Méi may treasure-and-receive it. Jiǎshēn [1704], 5th month, rénxū-day, (subject) Lǐ Guāngdì respectfully records.
Abstract
Composition window: c. 1670 (Méi Wéndǐng’s first surviving mathematical writings; he was approximately 37) – 1721 (his death; the latest works in the anthology). The composition span is therefore over fifty years, with most of the major works completed in the 1680s-1700s. Posthumous assembly: 1721–1726 (Wèi Lìtóng obtains the papers, Yáng Zuōméi edits-and-supplements). First Sìkù-recension reorganization: 1781.
The work’s significance:
(a) The foundational early-Qīng synthesis of Chinese-and-Western mathematical-astronomical learning: Méi Wéndǐng’s project — the systematic comparative engagement of the Chinese mathematical tradition (the Jiǔzhāng, the Zhōubì, the Shòushí) with the European mathematical tradition (Euclidean geometry, spherical trigonometry, Tychonic-Keplerian astronomy as transmitted by the Jesuits) — is the foundational achievement of high-Qīng mathematical historiography. Through the Lìsuàn quánshū and through Méi Juéchéng’s incorporation of Méi Wéndǐng’s work into the imperial KR3f0018 Lìxiàng kǎochéng and KR3f0019 Hòubiān, the Méi synthesis defined the Chinese mathematical curriculum for the eighteenth century.
(b) The Xīfǎ Zhōngyuán (Western methods originate from the East) framing: Méi Wéndǐng’s general historiographical position — that Western mathematical-astronomical superiority is real but represents the recovery and refinement of methods that had originally been Chinese (the Zhōubì’s implicit cosmology, the Jiǔzhāng’s implicit algebra, the Shòushí’s implicit precession-correction) — would become the standard high-Qīng historiographical position, repeated throughout the Sìkù tíyào (cf. KR3f0001, KR3f0014, KR3f0015, KR3f0019).
(c) The 1702 imperial encounter: Lǐ Guāngdì’s Gōngjì preface preserves one of the most vivid first-hand accounts of Kāngxī-period imperial-scholarly interaction. The Kāngxī emperor’s personal reading of Méi’s Lìxué yíwèn, his marginalia-and-comments, his face-to-face dialogue with Lǐ Guāngdì about the work’s merits-and-defects, and his ultimate verdict (“no defects, but the computational methods are not yet complete”) together constitute a unique documentary moment in late-imperial Chinese intellectual history. The episode — and the Jīxué cānwēi imperial calligraphy bestowed on Méi — established Méi’s reputation and made possible the broader institutional reception of his work through his grandson Méi Juéchéng.
(d) The 29-title structure as Chinese mathematical curriculum: the 29 sub-titles of the Lìsuàn quánshū — covering everything from elementary calendar-theory to advanced spherical trigonometry, from ancient Chinese problems (the Nine Chapters’ fāngtián, shǎoguǎng, gōugǔ) to introduced European techniques (Euclidean geometry, Napier’s-bones, spherical trigonometry, eclipse-prediction) — together constitute a comprehensive Chinese mathematical curriculum. The KāngxīYōngzhèng imperial mathematical academy’s Shùlǐ jīngyùn (1722) and the broader Qīng mathematical-pedagogical tradition all draw heavily on this curricular structure.
The 提要’s verdict — zì Guō Shǒujìng yǐlái hǎn jiàn qí bǐ “from Guō Shǒujìng onward, [his] equal has rarely been seen” — places Méi Wéndǐng in direct succession with the Yuán-period architect of the Shòushí lì, leapfrogging the entire Míng dynasty’s mathematical-astronomical production. This is one of the most laudatory editorial assessments in the entire Sìkù tíyào corpus.
For the related Méi-family works, see KR3f0027 Dàtǒng lìzhì, KR3f0028 Wù’ān lìsuàn shūjì, and KR3f0029 ZhōngXī jīngxīng tóngyì kǎo (by his brother Méi Wénmǐng). For the imperial works that absorbed his mathematical contributions, see KR3f0018 Lìxiàng kǎochéng and KR3f0019 Hòubiān. For the principal author, see 梅文鼎.
Translations and research
- Han Qi 韓琦, Tōng-tiān zhī xué: Yēsū-huì shì hé tiānwén-xué zài Zhōng-guó de chuán-bō 通天之學, Beijing: Sānlián, 2018 (treats Méi Wéndǐng extensively).
- Hashimoto Keizō 橋本敬造. Mei Wending’s Investigations into Calendrical Astronomy, in Historia Scientiarum (various issues).
- Jami, Catherine. The Emperor’s New Mathematics: Western Learning and Imperial Authority during the Kangxi Reign (1662–1722), Oxford: Oxford University Press, 2012 (essential context, with major treatment of Méi).
- Liu Dun 劉鈍, “Méi Wéndǐng zhī shù-xué 梅文鼎之數學” (various journal articles), Beijing: Kē-xué Chū-bǎn-shè.
- Cullen, Christopher. Heavenly Numbers: Astronomy and Authority in Early Imperial China, Oxford: Oxford University Press, 2017.
- Lǐ Yán 李儼, Zhōng-suàn-shǐ lùn-cóng 中算史論叢, Beijing: Kē-xué Chū-bǎn-shè, multiple volumes.
- Mei Rongzhao 梅榮照, Míng-Qīng shù-xué-shǐ lùn-wén jí 明清數學史論文集, Nánjīng: Jiāngsū Jiào-yù Chū-bǎn-shè, 1990.
- Engelfriet, Peter M. Euclid in China, Sinica Leidensia 40, Leiden: Brill, 1998.
Other points of interest
The Cáo Fǔ 曹頫 connection — the Kāngxī emperor’s specific commission of the imperial textile bondservant of the famous Cáo family of Sūzhōu (the family later famous as the model for Cáo Xuěqín’s Hónglóu mèng of c. 1750) to manage Méi Wéndǐng’s funeral arrangements — is one of the most personally-affecting moments in the late-imperial Chinese imperial-patronage record. The episode places Méi Wéndǐng directly in the social-and-cultural network that would shortly produce the greatest novel of the Qīng dynasty.
The 提要’s framing of Méi Wéndǐng as a cǎoyě shūshēng (rural village book-student) who nonetheless achieved imperial recognition — and surpassed even Guō Shǒujìng — is a paradigm of the high-Qīng meritocratic-of-intellect ideology. The implicit contrast with the institutionally-positioned Western Jesuits (who came as Bureau Directors) is part of the editorial framing: Méi Wéndǐng demonstrates that Chinese private scholarship can match-and-exceed the institutionally-supported foreign expertise.