Gōugǔ chǐ cèliáng xīnfǎ 勾股尺測量新法
New Methods of Surveying with the Right-Triangle Sighting-Rod by 劉衡 (撰)
About the work
劉衡 Liú Héng’s (1776–1841) treatise in 2 juàn on practical surveying using the gōugǔ chǐ 勾股尺 — a right-angled sighting-rod calibrated with arithmetic scales for the rapid graphical computation of the gōugǔ 勾股 (right-triangle) problem-types that occur in surveying.
Abstract
The instrument here called gōugǔ chǐ 勾股尺 is the right-triangle sighting-rod — an L-shaped graduated instrument used in indigenous Chinese surveying to determine distances and heights by sightings combined with similar-triangle proportions. The fundamental surveying procedure is the zhòngchā 重差 (“repeated difference”) method codified in the Hǎidǎo suànjīng 海島算經 of 劉徽 Liú Huī (KR3fc003’s parent text): two sightings at known horizontal separation give two similar right triangles, from which the unknown height or distance can be computed.
劉衡 Liú Héng’s 2-juàn treatise presents systematic procedures for surveying using a properly-graduated gōugǔ chǐ, with worked examples covering the standard problem-types: distance to an inaccessible object, height of a remote feature, depth of a valley, breadth of a river. The title’s xīnfǎ 新法 (“new methods”) indicates that the treatise presents Liú Héng’s own systematic working — likely an improvement on the indigenous zhòngchā procedures by incorporating scales calibrated for direct readings rather than computational reductions.
The work belongs to the active mid-Qīng tradition of practical surveying mathematics — alongside 李潢 Lǐ Huáng’s KR3fc003 Hǎidǎo suànjīng xìcǎo túshuō and the various Méi-school treatments. It is closely paired with the companion treatise KR3fc063 (sundial-laying with the chǐsuàn) as Liú Héng’s two principal works on practical instrument-mathematics.
Dating: Liú Héng lived 1776–1841. notBefore 1800; notAfter 1841.
Translations and research
- Cullen, Christopher. 2007. The Suàn shù shū 算數書 ‘Writings on Reckoning’: A Translation. Cambridge: Needham Research Institute. — Discusses the early zhòng-chā surveying tradition.
- Wú Wénjùn 吳文俊, ed. 1985. Zhōng-guó shù-xué shǐ dà-xì 中國數學史大系, vol. 7.