Yùzhì Shùlǐ jīngyùn 御製數理精蘊
Imperially Composed Essential Principles of Mathematics by 聖祖玄燁 (the Kāngxī emperor, 1654–1722, 清, chìzhuàn 敕撰 — by imperial decree composed); editorial direction by 允祿 Yǔnlù et al., with 何國宗 Hé Guózōng, Méi Juéchéng 梅㲄成, and others as principal mathematical staff. Chartered Kāngxī 52 (1713); printed Yōngzhèng 2 (1724).
About the work
The mathematical core of the great Kāngxī-Yōngzhèng-period imperial synthesis Lǜlì yuānyuán 律歷淵源 (Foundations of Music and Astronomy, 100 juàn total). In 53 juàn, the work systematically expounds the entire mathematical curriculum then accessible — combining the foundational Chinese mathematical tradition (the Hétú, Luòshū, Zhōubì jīng, Jiǔzhāng, suànfǎ) with the European mathematics transmitted through the Jesuit channel (Euclidean geometry, trigonometry, logarithms, algebra). The work is the foundational systematic mathematical curriculum of the high Qīng and was the working reference for Chinese mathematical education through the late-Qīng period.
Structure (per the 提要):
Upper Edition (上編 shàngbiān) — 5 juàn — Lìgāng míngtǐ 立綱明體 (Establishing the Outline, Clarifying the Body):
- Shùlǐ běnyuán 數理本源 (Source-and-Foundation of Numerical Principle)
- Hétú 河圖 (Yellow-River Diagram)
- Luòshū 洛書 (Luò-River Writing)
- Zhōubì jīng jiě 周髀經解 (Explanation of the Zhōubì)
- Jǐhé yuánběn 幾何原本 (Euclid’s Elements — re-exposition)
- Suànfǎ yuánběn 算法原本 (Original-Foundations of Calculation Methods)
Lower Edition (下編 xiàbiān) — 40 juàn — Fēntiáo zhìyòng 分條致用 (Categorical Practical Application):
- Shǒubù 首部 (Head Section — basic operations)
- Xiànbù 線部 (Line Section — linear and proportional procedures)
- Miànbù 面部 (Surface Section — area calculations)
- Tǐbù 體部 (Body Section — solid-volume calculations)
- Mòbù 末部 (End Section — including jiègēnfāng 借根方 algebra)
Tables (表 biǎo) — 8 juàn:
- Bāxiàn biǎo 八線表 (Eight-Lines Table — trigonometric values: sin, cos, tan, cot, sec, csc, vers, covers)
- Duìshù chǎnwēi biǎo 對數闡微表 (Subtle-Exposition Logarithm Table)
- Duìshù biǎo 對數表 (Logarithm Table)
- Bāxiàn duìshù biǎo 八線對數表 (Eight-Lines-Logarithm Table)
The 提要’s substantive comparative-mathematical observations:
(a) The reform of fāngchéng: the old Chinese fāngchéng (system-of-equations) tradition had treated the 2-color, 3-color, 4-color, 5-color cases as separate methods, with substantial methodological errors in the positive-and-negative-number rules. The Shùlǐ jīngyùn unifies all cases into four general patterns (héshù 和數 — sum-numbers; jiàoshù 較數 — difference-numbers; héjiào jiānyòng — combined sum-and-difference; héjiào jiāobiàn — sum-and-difference exchange-variation), simplifying the apparatus and correcting the errors. This is the foundational systematization of Gaussian-style elimination in the Chinese tradition.
(b) The π refinement: the work computes π by both Zǔ Chōngzhī’s 6-sided-inscribed-polygon method and Zhào Yǒuqīn’s 4-sided-inscribed-polygon method (KR3f0005), showing that both yield the same value 3.14159625, equivalent to the Western value. The work then reforms the húshǐ (arc-and-arrow) procedures to use this refined value, replacing the inadequate older Chinese jìngyī wéisān 徑一圍三 (3:1 ratio) approximation.
(c) The trigonometric tables and the eight-lines: the work introduces the bāxiàn 八線 (eight lines) — the eight trigonometric functions sin, cos, tan, cot, sec, csc, versin, coversin — at intervals of 1 cùn / 10 miǎo across one xiàngxiàn (quadrant), tabulating to high precision.
(d) Extension of Euclidean solid geometry: the work supplements the 6-book Ricci-Xú KR3f0047 Jǐhé yuánběn with the Euclidean material on regular solids (treated in the original Elements books XIII and XV but never translated by Ricci-Xú). The Shùlǐ jīngyùn computes volumes of inscribed-and-circumscribed regular polyhedra up to 12-and-20-sided, using the lǐfēnzhōngmò xiàn 理分中末線 (the golden ratio — median-and-extreme proportion of Euclid’s book VI definition 3).
(e) Algebra (jiègēnfāng 借根方): the work introduces the European algebra received through the Jesuit channel. The 提要 explicitly identifies jiègēnfāng as the recovered lì tiānyuányī 立天元一 of Lǐ Yě (KR3f0042) and Qín Jiǔsháo (KR3f0041) — the Xīfǎ Zhōngyuán 西法中源 framing.
(f) Logarithms and proportional compass: the work introduces logarithms (received via 薛鳳祚 Xuē Fèngzuò’s KR3f0024 KR3f0025 Smogulecki transmission) and proportional-compass methodology (bǐlìguī jiě 比例規解), framing both as “calculation by quantity, replacing computation by computation”.
The 提要’s verdict — for the entire work and for the Kāngxī project as a whole — is exceptional: “[the work is] indeed an unprecedented book; even specialist named-houses cannot peep at one part-in-ten-thousand of its height-and-depth”. This is one of the most enthusiastic editorial verdicts in the entire Sìkù tíyào corpus.
For the parallel Lǜlì yuānyuán parts, see KR3f0018 Lìxiàng kǎochéng. For the principal author, see 聖祖玄燁. For the principal Chinese-mathematical-staff figures, see 允祿, 何國宗, 梅文鼎, 李冶 (recovered through KR3f0042), 秦九韶 (recovered through KR3f0041). For the European mathematical sources, see 利瑪竇 (Ricci) and the various Jesuit collaborators of the Kāngxī mathematical academy.
Tiyao
[Full text summarized above. Dated Qiánlóng 46 (1781), ninth month.]
Translations and research
- Jami, Catherine. The Emperor’s New Mathematics: Western Learning and Imperial Authority during the Kangxi Reign (1662–1722), Oxford: Oxford University Press, 2012. The standard study.
- Han Qi 韓琦, Tōng-tiān zhī xué 通天之學, Beijing: Sānlián, 2018.
- Hashimoto Keizō 橋本敬造. Hō Yū-ran, Kyoto: Kansai University Press, 1988.
- Standaert, Nicolas (ed.). Handbook of Christianity in China, vol. 1, Leiden: Brill, 2001.
- Liu Dun 劉鈍, Yǒu-fēng-yǒu-yún de Shù-lǐ jīng-yùn (有風有雲的數理精蘊 — various journal articles).
Other points of interest
The Kāngxī mathematical academy (the Méngyǎng zhāi 蒙養齋, established Kāngxī 52 / 1713 in the inner palace) is among the most institutionally important developments of the high Qīng. Through the Méngyǎng zhāi the Kāngxī emperor created a permanent imperial mathematical-and-astronomical research-and-teaching establishment, descendant of which (through the Yōngzhèng-Qián-lóng-period continuations) would constitute the institutional thread of imperial Chinese mathematical research from the late Míng to the late Qīng.
The 提要’s Xīfǎ Zhōngyuán identification of jiègēnfāng (algebra) with lì tiānyuányī — anchored in Méi Juéchéng’s Chìshuǐ yízhēn and articulated authoritatively in this 提要 — is the foundational late-imperial Chinese statement of the SòngYuán recovered thesis. Through this framing, the 18th-century Chinese mathematical community could integrate the imported Western algebra into the Chinese mathematical canon as the rediscovery of native methodology, rather than as a foreign import.