Chóubiǎo kāi zhūchéngfāng jiéfǎ 籌表開諸乘方捷法
A Rapid Tabular Method for the Extraction of n-th Roots by 劉衡 (撰)
About the work
劉衡 Liú Héng’s (1776–1841) treatise in 4 juàn on a tabular jiéfǎ 捷法 (“rapid method”) for the extraction of arbitrary n-th roots (zhūchéngfāng 諸乘方, literally “various multiplications-of-squares,” the indigenous Chinese label for higher-degree powers). The method uses precomputed chóubiǎo 籌表 (“rod-tables”) — printed tables of selected powers — to reduce the root-extraction procedure to a sequence of table look-ups.
Abstract
The classical Chinese root-extraction procedure is the zēngchéng kāifāng 增乘開方 algorithm of the SòngYuán tradition — the Horner-style synthetic-division method developed by 賈憲 Jiǎ Xiàn and 秦九韶 Qín Jiǔsháo and preserved in the Shùshū jiǔzhāng (KR3fc008). The procedure is fully general (it works for any-degree polynomial root extraction) but computationally laborious: each digit of the root requires a full pass through the coefficient sequence.
Liú Héng’s Chóubiǎo jiéfǎ aims to reduce this labour for the special case of pure n-th root extraction (xⁿ = N for given N and n). The procedure precomputes a table of n-th powers of carefully chosen numbers and reduces the extraction to: (1) bracketing N between consecutive tabulated powers, (2) interpolating to obtain the leading digits of the root, (3) applying a correction procedure for the remaining digits. The 4-juàn presentation lays out the tables, the procedure, and an extensive collection of worked examples covering n from 2 (ordinary square roots) up to n = 10 or thereabouts.
The work prefigures the late-Qīng / Tóngzhì interest in computational tabulation that would culminate in 戴煦 Dài Xù’s KR3fc073 Qiúbiǎo jiéshù 求表捷術 (rapid methods for table-construction, especially trigonometric and logarithmic tables) and 李善蘭 Lǐ Shànlán’s translation work on logarithms. Liú Héng’s tabular root-extraction is the indigenous-tradition precursor of these mid-nineteenth-century tabular projects.
Dating: Liú Héng lived 1776–1841. notBefore 1800; notAfter 1841.
Translations and research
- Tián Miǎo 田淼. 2003. Zhōng-guó shù-xué de xī-huà lì-chéng 中國數學的西化歷程. Jǐ-nán: Shān-dōng jiāo-yù chū-bǎn-shè.
- Wú Wénjùn 吳文俊, ed. 1985. Zhōng-guó shù-xué shǐ dà-xì 中國數學史大系, vol. 7–8.