測圓海鏡細草

Cèyuán hǎijìng xìcǎo 測圓海鏡細草

Detailed Procedural Exposition of the Sea Mirror of Circle Measurement by 李冶 (撰)

About the work

A non-WYG recension of 李冶 Lǐ Yě’s Cèyuán hǎijìng 測圓海鏡 (1248) — his magnum opus on the geometry of right-triangles inscribed in circles using the lì tiānyuányī 立天元一 algebraic-equation method — preserved here under the title Cèyuán hǎijìng xìcǎo (with the xìcǎo “detailed-procedure” notation by Lǐ Yě himself as integral part of the text). The WYG companion is KR3f0042. The catalog entry is annotated “Mathematik basierend auf Song” (German: “mathematics based on Sòng”) in the meta — a note by an earlier cataloger emphasizing that Lǐ Yě’s work belongs intellectually to the Sòng mathematical tradition, even though Lǐ Yě himself died in the Yuán.

Abstract

The work, in 12 juàn in the WYG recension or 16 juàn in the present KX recension (the discrepancy reflects sub-juàn divisions), contains 170 problems on the geometry of gōugǔ róngyuán 勾股容圓 (“right-triangle containing a circle”) configurations — specifically, the metric relations among the radius of an inscribed circle, the sides of the containing right triangle, and the line-segments produced by tangency points. Lǐ Yě systematizes the topic by working out, for each of fifteen distinct geometric configurations (the fánlì 凡例 categorization at the beginning of the work), the full set of metric relations that can be derived; the 170 problems represent the systematic exhaustion of these configurations.

The work’s principal mathematical-historical contribution is its systematic use of the lì tiānyuányī 立天元一 (“setting up the heavenly origin as one”) algebraic-equation method. The method introduces a notational system in which one of the unknown quantities is designated yuán 元 (“origin”) and represented by the marker yuán 元 placed alongside the column for the corresponding power; polynomial expressions are written as columns of counting-rod numerals with the row indicating the power, and equations are formed by setting two such polynomial expressions equal. This is in effect the first systematic polynomial-equation notation in any tradition, allowing complex algebraic manipulation by direct symbolic operation on the rod-arrangements. Lǐ Yě’s elaboration of the method in 170 worked problems is the most thoroughgoing in the SòngYuán mathematical tradition; without his work (and 秦九韶 Qín Jiǔsháo’s KR3f0041 parallel use of the method), the tiānyuányī would probably not have survived into the late-imperial period.

For full biographical and mathematical discussion of Lǐ Yě — his Jīn jìnshì of 1230, his refusal of repeated Yuán imperial summons, the famous late-Míng forgetting of the tiānyuányī method documented by 顧應祥 Gù Yìngxiáng (see KR3f0043), and the Kāng-xī-period recovery via Méi Juéchéng’s identification of tiānyuányī with European jiègēnfāng — see the person note at 李冶 and the WYG entry at KR3f0042.

The textual transmission of the Cèyuán hǎijìng is unusually complete by the standards of SòngYuán mathematical works: Lǐ Yě’s own preface dates the work Yuánzhèng 5 (1248); the SòngYuán transmission remained continuous through the early Míng, with multiple Míng manuscript and print witnesses; the Sìkù-period collation was therefore unproblematic. The present KX recension preserves the 16-juàn division of the older Míng line, distinct from the WYG 12-juàn division but with substantively identical content. Dating: 1248 per Lǐ Yě’s preface.

Translations and research

• Chemla, Karine. 1982. Étude du livre Reflets des mesures du cercle sur la mer de Li Ye. Doctoral dissertation, University of Paris. 4 vols. — Comprehensive French study with full translation; the principal Western-language scholarly treatment.
• Hé Bǐngyù 何丙郁 (John Hoe). 1977. Les systèmes d’équations polynomes dans le Siyuan yujian (1303). Paris: Collège de France. — Discusses Lǐ Yě’s tiān-yuán-yī as the precursor to Zhū Shìjié’s sì-yuán.
• Martzloff, Jean-Claude. 1997. A History of Chinese Mathematics. Berlin: Springer. — Includes detailed analysis of Lǐ Yě’s algebraic methods.
• Lam Lay-Yong and Ang Tian-Se. 2004. Fleeting Footsteps, rev. edn. Singapore: World Scientific.

Links

• WYG companion: KR3f0042
• Sister work by same author: KR3f0044 Yìgǔ yǎnduàn
• Late-Míng partial commentary: KR3f0043 Cèyuán hǎijìng fēnlèi shìshù (Gù Yìngxiáng)
• Wikipedia: https://en.wikipedia.org/wiki/Ceyuan_haijing