Gōugǔ suànshù 勾股算術
Mathematical Methods of the Right Triangle by 顧應祥 (撰)
About the work
A mid-Míng mathematical treatise on the right triangle in 3 juàn by 顧應祥 Gù Yìngxiáng (1483–1565), independent of but complementary to his Cèyuán hǎijìng fēnlèi shìshù (KR3f0043) and Húshǐ suànshù (KR3f0045). Composed in the closing decades of Gù Yìngxiáng’s life (after his retirement from active office in the 1540s).
Abstract
Gù Yìngxiáng’s Gōugǔ suànshù is one of the principal late-Míng treatments of the gōugǔ (right-triangle) chapter of the KR3f0032 Jiǔzhāng tradition. The work systematizes the metric relations among the three sides of a right triangle and their related quantities (sum-and-difference of sides, area, inscribed circle), presenting them as a closed system of computational procedures.
Like Gù Yìngxiáng’s better-known Cèyuán hǎijìng fēnlèi shìshù, this work avoids the tiānyuányī 立天元一 algebraic-equation method that the SòngYuán mathematicians had used to derive these relations. The Sìkù 提要 of KR3f0043 explicitly documents Gù Yìngxiáng’s failure to understand tiānyuányī and his consequent removal of the method from his expositions — a critique that applies to the Gōugǔ suànshù as well. The work therefore proceeds by elementary computational and proportional methods, presenting each relation as a stand-alone procedural recipe rather than as a derivation from a unified algebraic foundation.
The work is nevertheless of considerable documentary value. Gù Yìngxiáng’s elementary-computational expositions found readers who would not have engaged with Lǐ Yě’s more demanding original, and through the Gōugǔ suànshù the right-triangle problem-types of the SòngYuán tradition continued to circulate in late-Míng practical-mathematical writing. The work also preserves a number of right-triangle problem-types that subsequent abacus-arithmetic works (notably 程大位 Suànfǎ tǒngzōng) would absorb into the standard problem-collection.
The work is also one of the principal pre-Jesuit late-Míng demonstrations of the limits of indigenous Chinese mathematical knowledge of the right triangle: where Gù Yìngxiáng works problem-by-problem, 徐光啟 Xú Guāngqǐ’s roughly contemporary KR3fc026 Gōugǔ yì (1606) introduces the Euclidean demonstration-and-proof methodology imported through the Jesuit translation of the Elements — a methodological contrast that exemplifies the late-Míng transition from indigenous procedural mathematics to imported demonstrative mathematics.
Dating: NotBefore 1540 (after Gù Yìngxiáng’s retirement from official service); notAfter 1565 (his death year).
Translations and research
- Engelfriet, Peter M. 1998. Euclid in China: The Genesis of the First Chinese Translation of Euclid’s Elements, Books I–VI (Jihe yuanben; Beijing, 1607) and Its Reception up to 1723. Leiden: Brill. — Provides the comparative context of Gù Yìngxiáng’s gōu-gǔ work against the Jesuit-introduced Euclidean methodology.
- Hé Bǐngyù 何丙郁 (John Hoe). 1977. Les systèmes d’équations polynomes dans le Siyuan yujian (1303). Paris: Collège de France. — Documents the late-Míng forgetting of tiān-yuán-yī.
- Wú Wénjùn 吳文俊, ed. 1985. Zhōng-guó shù-xué shǐ dà-xì 中國數學史大系, vol. 6.
- Martzloff, Jean-Claude. 1997. A History of Chinese Mathematics. Berlin: Springer.