Yìgǔ yǎnduàn 益古演段
Elaborated-Sectional Exposition of [the Method-Book] Yì-gǔ by 李冶 (Lǐ Yě, 1192–1279, 元, zhuàn 撰), composed Zhìyuán rénwǔ (1282 — but Lǐ Yě died 1279, so the date here cannot be the original composition; the 1282 date refers to the printing per Yàn Jiān’s 硯堅 preface). Internal evidence (Yàn Jiān’s preface, which says the work was prepared after Lǐ Yě’s KR3f0042 Cèyuán hǎijìng was already in circulation) places the composition between 1248 (the Cèyuán hǎijìng completion) and 1279 (Lǐ Yě’s death). A specific composition date of c. 1259 has been suggested by modern scholars.
About the work
Lǐ Yě’s 3-juan elementary-pedagogical work designed to make the lì tiānyuányī 立天元一 algebraic methodology accessible to beginners. The work re-works a now-lost earlier work — the Yìgǔ jí 益古集 by an unknown earlier author (Lǐ Yě’s preface mentions only “近世有某 — recent generation has [some] [Mr.] So-and-So”) — into 64 problems with full pedagogical exposition: each problem includes cǎo (procedural-elaboration showing the tiānyuányī algebraic setup), tiáoduàn (sectional-decomposition showing how the Jiǔzhāng’s elementary methods (fāngtián, shǎoguǎng) relate to the algebraic procedure), tú (geometric diagrams illustrating the underlying spatial reasoning), and yì (meaning-explanation in plain prose).
The work’s pedagogical strategy:
(I) Begin with circles-and-squares: Lǐ Yě restricts the problem-set to circle-and-square area-and-perimeter relations because (per the 提要 paraphrasing Lǐ Yě) “although [ lì tiānyuányī is the root of all the methods, [its] divine-and-changing-and-transforming has no end; the student suddenly wishing to penetrate [it], [finds] vast-vastly without doorway-or-path of entry. Only by means of square-and-circle areas to clarify [it]; the principle is then easily seen”.
(II) Each circle-and-square problem includes the cǎo-procedure: this is precisely the section that Gù Yìngxiáng’s KR3f0043 would later remove. Lǐ Yě intended the cǎo-procedures as the core of the pedagogical content.
(III) The pedagogical purpose is the recovery of method-from-secrecy: Lǐ Yě’s preface (preserved in the 提要 quotation) is sharply polemical against the late-Sòng / early-Yuán mathematical guild’s secrecy-culture: “the present-age mathematicians are not necessarily of the workmanship of [Liú] [Huī] and Lǐ [Chúnfēng], [yet] their narrow-minds-and-stooped-vision are not willing to clearly display [methods] to others — only working at yǐnhù cuòróu (hiding-mutual-and-twisting-confused), so as to make [matters] míngxìng àndǎn (gloomy-vague and dark-shadowy), only afraid that students might get to peek at the resemblance”. Lǐ Yě’s stated motivation is the anti-secrecy democratization of mathematical knowledge.
The work was nearly lost in the Míng (per the 提要); only the Yǒnglè dàdiǎn preserved a complete copy. Gù Yìngxiáng and Táng Shùnzhī, who had the main Cèyuán hǎijìng but could not understand the tiānyuányī method, did not have access to the more pedagogical Yìgǔ yǎnduàn — which would have provided exactly the methodological introduction they needed.
The Sìkù 提要 specifically commends the work as the gōuniè 圭臬 (compass-and-gnomon, meta-standard) of the calculation-school: it is the work through which the lì tiānyuányī method should be approached pedagogically. The Sìkù-recension is recovered from the Yǒnglè dàdiǎn and reorganized into 3 juàn per Yàn Jiān’s preface (the Yǒnglè dàdiǎn did not preserve the juàn-divisions).
For the related works by Lǐ Yě, see KR3f0042 Cèyuán hǎijìng (his magnum opus that this work prepares the student for). For the failed Míng-period exposition, see KR3f0043 Cèyuán hǎijìng fēnlèi shìshù. For the principal author, see 李冶. For the foundational Sòng tiānyuányī source, see KR3f0041.
Tiyao
[Full 提要 text in source file. Key points already summarized above. Dated Qiánlóng 46 (1781), seventh 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 both Lǐ Yě works).
- Martzloff, Jean-Claude. A History of Chinese Mathematics, Berlin: Springer, 1997.
- Libbrecht, Ulrich. Chinese Mathematics in the Thirteenth Century, Cambridge MA: MIT Press, 1973.
Other points of interest
Lǐ Yě’s preface’s polemic against mathematical-guild secrecy is one of the most explicit pre-modern Chinese statements of the open-knowledge ideal in scientific scholarship. The contrast with the late-Míng / early-Qīng jiāxué (family-school) tradition of mathematical secrecy (where mathematical methods were transmitted privately within hereditary specialist families) makes Lǐ Yě’s anti-secrecy posture particularly remarkable.
The Yàn Jiān 硯堅 1282 preface preserves the most detailed account of the work’s compositional context: Lǐ Yě’s friend Lǐ Shīzhèng 李師徵’s brother Lǐ Shīguī 李師珪 commissioned the printing after Lǐ Yě’s Cèyuán hǎijìng was already in circulation. The personal-network detail anchors the work’s transmission in Lǐ Yě’s late-life social circle in northern China during the Mongol consolidation.