Kāi zhūchéngfāng jiéshù 開諸乘方捷術
A Rapid Procedure for the Extraction of n-th Roots by 項名達 (撰)
About the work
項名達 Xiàng Míngdá’s (1789–1850) treatise on a rapid procedure (jiéshù 捷術) for the extraction of arbitrary n-th roots (zhūchéngfāng 諸乘方, the indigenous Chinese term for general higher-degree powers). The work belongs to the same problem-domain as 劉衡 Liú Héng’s KR3fc065 Chóubiǎo kāizhūchéngfāng jiéfǎ and is a key procedural-mathematical document of the early-Dào-guāng generation.
Abstract
The extraction of n-th roots — given N and n, find x such that xⁿ = N — is one of the central computational problems of the Chinese mathematical tradition. The classical procedure is the zēngchéng kāifāng 增乘開方 Horner-style synthetic-division algorithm developed in the SòngYuán period and preserved in 秦九韶 Qín Jiǔsháo’s KR3fc008 Shùshū jiǔzhāng. The procedure is fully general but laborious for high n and large N.
Xiàng Míngdá’s Kāi zhūchéngfāng jiéshù presents a procedural improvement, exploiting the binomial-expansion structure of (a+b)ⁿ to organise the extraction more efficiently. Where 劉衡 Liú Héng’s KR3fc065 approach relies on precomputed tables, Xiàng Míngdá’s approach is purely algorithmic — a reorganisation of the computation rather than a tabulation of intermediate values. The two approaches were the principal competing methods for rapid root extraction in mid-Qīng mathematical practice.
The work was probably composed in the same general period as Xiàng’s KR3fc070 Gōugǔ liùshù and KR3fc071 Sānjiǎo héjiào shù — the mature productive years of the 1830s–1840s. Internal evidence suggests connections with the cyclotomic series-expansion work in KR3fc069 Xiàngshù yīyuán, where the binomial expansion plays a similar foundational role.
Dating: notBefore 1820; notAfter 1850. (The catalog gives only Qing dynasty; tighter bracket adopted on the basis of Xiàng’s lifedates and the connection to his other works.)
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.