Cèyuán hǎijìng 測圓海鏡
Sea-Mirror of Circle Measurements by 李冶 (Lǐ Yě, 1192–1279, 元, zhuàn 撰), composed Mìngchāng 5 = Chúnyòu 8 (1248)
About the work
Lǐ Yě’s monumental 12-juan magnum opus on the geometry of right-triangles inscribed in circles, completed 1248. The work systematizes the gōugǔ róngyuán 勾股容圓 (right-triangle-containing-circle) configuration: 15 distinct geometric figures derived from variations of a basic right-triangle with inscribed and circumscribed circles, with each figure giving rise to multiple measurable line segments and areas. The work then presents 170 problems based on these configurations, each problem progressing from the geometric description through the cǎo 草 (procedural-elaboration) using lì tiānyuányī 立天元一 (algebraic-variable methodology) to the final answer.
The work’s foundational mathematical contribution is the systematic elaboration of lì tiānyuányī — the algebraic-equation methodology developed by Qín Jiǔsháo in KR3f0041 Shùshū jiǔzhāng (1247, just one year before Lǐ Yě’s Cèyuán hǎijìng). Lǐ Yě’s contribution: where Qín Jiǔsháo had used lì tiānyuányī as one specialized technique within his much broader 9-chapter compendium, Lǐ Yě developed the method as the principal-and-systematic procedure for an entire class of geometric problems. The 170 problems of the Cèyuán hǎijìng are essentially 170 demonstrations of lì tiānyuányī applied to right-triangle-circle configurations — making the work the foundational systematic exposition of SòngYuán algebraic methodology.
The work’s structural organization (per the 提要):
(I) Identification of the 15 basic figures: from a basic right-triangle with inscribed circle (gōugǔ róngyuán), variations are produced by considering the same configuration from different sighting points (the circle’s center, points outside the circle, the triangle’s vertex, etc.) — yielding 15 distinct geometric figures, each with its own characteristic measurable quantities.
(II) Several hundred labeled-and-marked relations (shíbié zájì shùbǎi tiáo 識别雜記數百條): for each figure, the work catalogs the geometric relations (sum-and-difference, product-and-quotient) among the labeled measurements.
(III) 170 problems: each problem specifies a particular figure and certain measurable values, and asks for derived values — with the cǎo (procedural-elaboration) showing how lì tiānyuányī sets up an algebraic equation in the unknown and solves it.
The Sìkù 提要 articulates a celebrated historical-mathematical narrative around the work:
(a) SòngYuán transmission: the lì tiānyuányī method appeared first in KR3f0041 Qín Jiǔsháo’s Dàyǎn chapter; was elaborated by Lǐ Yě here; further developed in the Yuán Shòushí lì’s 草 (procedural-elaboration); and definitively presented in Zhū Shìjié’s 朱世傑 Sìyuán yùjiàn 四元玉鑑 of 1303.
(b) Late-Míng forgetting: the method was forgotten by the late Míng. Táng Shùnzhī 唐順之 wrote to Gù Yìngxiáng 顧應祥 saying lì tiānyuányī “absolutely means nothing to me” (màn bù shěng wéi hé yǔ 漫不省為何語). Gù Yìngxiáng’s KR3f0043 Cèyuán hǎijìng fēnlèi shìshù attempted to expound Lǐ Yě’s work but had to remove the tiānyuányī sections because Gù himself could not understand them.
(c) The 1610 Jesuit gap: Matteo Ricci, Xú Guāngqǐ, and Lǐ Zhīzǎo translated the Tóngwén suànzhǐ 同文算指 (KR3f0046) and other Western mathematical works, but specifically did not explain lì tiānyuányī. The 提要 cites Xú Guāngqǐ’s KR3f0014 Gōugǔ yì preface admission that he wished to expound Cèyuán hǎijìng’s meaning “but had not yet had leisure” — i.e., he had not understood it.
(d) Kāngxī-period recovery via European algebra: when European algebra (jiègēnfāng 借根方, Algebra) was transmitted to the Kāngxī court (probably via Antoine Thomas, S.J., in the 1690s), Méi Juéchéng 梅㲄成 (Méi Wéndǐng’s grandson, working at the Méngyǎng zhāi) recognized that this method was equivalent to the lost Lǐ Yě / Qín Jiǔsháo lì tiānyuányī. He documented this recognition in his Chìshuǐ yízhēn 赤水遺珍 with the European-language etymology Āěrrèbālā 阿爾熱巴拉 (Algebra) glossed as Dōnglái fǎ 東來法 (Eastern-Coming Method). Through this recognition the lost SòngYuán algebra was recoverable; the Sìkù 提要 explicitly endorses this reading.
The 提要’s closing statement is the foundational Sìkù statement of Xīfǎ Zhōngyuán applied to algebra: “Now using [the recovered Lǐ Yě method] to verify-and-check the Western method, [they are] in everything precisely-fitting; what [Méi] Juéchéng said is reliable-and-evidenced. Specifically [we] preserve [it] in record as the secret-key of the calculation methods, and to display that the Chinese method and the Western method mutually-illuminate-and-clarify each other — [there is] no need to set up boundaries-of-the-realm-of-views ( ménhù zhī jiàn )“.
For Lǐ Yě’s biography, see 李冶. For the related works by Lǐ Yě, see KR3f0044 Yìgǔ yǎnduàn and KR3f0043 (Gù Yìngxiáng’s failed Míng-period exposition). For the foundational Sòng tiānyuányī source, see KR3f0041 Shùshū jiǔzhāng. For the Wàn-lì-period Jesuit mathematical material that did not transmit tiānyuányī, see KR3f0014 Cèliáng fǎyì and KR3f0046 Tóngwén suànzhǐ.
Tiyao
[Full 提要 text in source file. Key points already summarized above. Dated Qiánlóng 46 (1781), second month.]
Translations and research
- 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 Chūbǎnshè, 1990 (treats the Lǐ Yě tradition extensively).
- Martzloff, Jean-Claude. A History of Chinese Mathematics, Berlin: Springer, 1997.
- Libbrecht, Ulrich. Chinese Mathematics in the Thirteenth Century, Cambridge MA: MIT Press, 1973 (treats Sòng-Yuán tiān-yuán-yī tradition).
- Lam Lay Yong and Ang Tian Se. Fleeting Footsteps, rev. ed., Singapore: World Scientific, 2004.
- Needham, Joseph (with Wang Ling), Science and Civilisation in China, vol. 3.
- Cullen, Christopher. Heavenly Numbers, Oxford: Oxford University Press, 2017.
Other points of interest
The 提要’s narrative arc — SòngYuán innovation → Míng-period forgetting → Kāngxī-period recovery via European mediation → recognition of identity through the European Catholic mathematical tradition — is one of the most consequential intellectual-historical narratives in the entire Sìkù corpus. It established the 高 modern academic genealogy of SòngYuán algebra through the jiègēnfāng / Āěrrèbālā identification.
The 提要’s specific etymological claim that Algebra (Āěrrèbālā) means Dōnglái fǎ 東來法 (Eastern-Coming Method) is — like the parallel claim about the Zhōubì 周髀 in KR3f0001 — a characteristic Xīfǎ Zhōngyuán assertion. The actual Arabic etymology of “algebra” (al-jabr) refers to the operation of re-uniting [broken parts] — specifically, the operation of moving a negative term from one side of an equation to the other where it becomes positive. There is no etymological connection to “the East”. The Méi Juéchéng-Sìkù-editorial framing here is mathematical-political rather than historically accurate.