Gōugǔ liùshù 勾股六術
The Six Procedures of Right-Triangle Mathematics by 項名達 (撰)
About the work
項名達 Xiàng Míngdá’s (1789–1850) treatise in 2 juàn presenting six systematic procedures (liùshù 六術) for the gōugǔ 勾股 (right-triangle) problem-types — the indigenous-Chinese counterpart of plane right-triangle trigonometry.
Abstract
The gōugǔ problem-domain is the indigenous Chinese formulation of the Pythagorean theorem and its associated problems: given two of {gōu 勾 (shorter leg), gǔ 股 (longer leg), xián 弦 (hypotenuse), area, sum-of-two-sides, difference-of-two-sides}, find the remaining elements. The Jiǔzhāng suànshù Chapter 9 (Gōugǔ) presents a representative set of problem-types but does not exhaust the combinatorial possibilities.
Xiàng Míngdá’s Gōugǔ liùshù presents six systematic procedures that together solve the full combinatorial space of gōugǔ problem-types. The “six procedures” are an organised classification: the first treats problems given two of the three side-lengths; the second treats problems involving one side plus area or perimeter; the third extends to problems involving sums-or-differences of sides; the fourth, fifth, and sixth handle the harder cases requiring two-step procedures or the use of additional auxiliary quantities. The presentation is more systematic than 李銳 Lǐ Ruì’s KR3fc058 Gōugǔ suànshù xìcǎo, which had focused on worked examples rather than systematic classification of problem-types.
The work belongs to Xiàng Míngdá’s reformulation programme — systematic reorganisation of the indigenous Chinese mathematical problem-types into compact, exhaustive procedural classifications. The same approach is applied to triangle geometry generally in his KR3fc071 Sānjiǎo héjiào shù.
Dating: Xiàng Míngdá lived 1789–1850. notBefore 1820 (mature productive period); notAfter 1850.
Translations and research
- Wú Wénjùn 吳文俊, ed. 1985. Zhōng-guó shù-xué shǐ dà-xì 中國數學史大系, vol. 7–8.
- Tián Miǎo 田淼. 2003. Zhōng-guó shù-xué de xī-huà lì-chéng 中國數學的西化歷程.