聞分館君 Ask BunKanKun

緝古算經考注

4 min read

Text ID
KR3fc005
Date
1800–1820
Dynasty
清

Jígǔ suànjīng kǎozhù 緝古算經考注

Investigative Annotation of the Continuing-the-Ancients Mathematical Canon by 李潢 (撰), 沈欽裴 (校訂)

About the work

The third member of 李潢 Lǐ Huáng’s triad of late-Qián-lóng / early-Jiā-qìng commentaries on the Suànjīng shíshū — joining the KR3fc002 Jiǔzhāng suànshù xìcǎo túshuō and the KR3fc003 Hǎidǎo suànjīng xìcǎo túshuō — this work is a 4-juàn detailed exposition and emendation of 王孝通 Wáng Xiàotōng’s Táng Jígǔ suànjīng 緝古算經 (KR3f0040), the principal cubic-equation treatise of pre-modern Chinese mathematics. Like the other two members of the triad, it was incomplete at Lǐ Huáng’s death in 1811 and was put through final editing and publication by 沈欽裴 Shěn Qīnpéi around 1820. (Note: the catalog records the author of this entry as the parent-text author Wáng Xiàotōng with Táng dynasty, but this is a meta-error — the work itself is Lǐ Huáng’s Qīng kǎozhù. The dynasty and persons here reflect the actual composer.)

Abstract

The Jígǔ suànjīng is the most mathematically demanding of the Táng-canonized Suànjīng shíshū: of its 20 problems, the second through the fourteenth all reduce to the formulation and numerical solution of cubic equations of the form x³ + ax² + bx = c (the standard pre-modern Chinese form, with the constant term on the right and the cubic and lower-order terms on the left, signs assumed positive). Wáng Xiàotōng’s procedural recipe in each case is geometric: the problem is reduced by area-and-volume analysis to a stated cubic, and the cubic is then solved by the kāi lìfāng 開立方 (cube-root extraction by Horner-style iteration). But the procedural reasoning is highly compressed and the geometric reduction is left implicit, with the result that by the late Qīng the work had become essentially unreadable to working mathematicians without specialized commentary.

Lǐ Huáng’s task was therefore expository in the strict sense: to write out, problem by problem, the implicit geometric reduction by which the cubic is generated; to supply the túshuō 圖說 (figure-and-explanation) for each reduction; and to work through the xìcǎo 細草 (detailed computation) of the cube-root extraction. In doing so Lǐ also identified and emended a number of textual corruptions in the received Jígǔ — particularly in the second juàn dealing with right-triangle-cubic problems, where several Sòng-era textual transpositions had rendered Wáng Xiàotōng’s procedure incoherent. Lǐ’s emendations were largely incorporated into the standard modern editions, particularly Qián Bǎocóng’s 1963 Suànjīng shíshū critical edition.

The work is also of broader historiographic significance as a benchmark of late-Qīng Chinese understanding of cubic-equation methods. It shows that by Lǐ Huáng’s generation, despite the importation of European jiègēnfāng 借根方 (borrowed-root) algebra under the Kāngxī court, the older indigenous kāifāng 開方 tradition of Wáng Xiàotōng and the SòngYuán algebraists had been fully recovered as a working method. This recovery was part of the same QiánJiā evidential project, driven also by 焦循 Jiāo Xún’s KR3fc048 Kāifāng tōngshì and the broader work of 李銳 Lǐ Ruì, that re-established the tiānyuányī and related algebraic methods as living mathematical practice rather than antiquarian curiosity.

Translations and research

  • Lam Lay-Yong. 1970. “The Geometrical Basis of the Ancient Chinese Square-Root Method.” Isis 61.1: 92–102. — Discusses Wáng Xiàotōng’s geometric cubic methods in the context of pre-modern Chinese root-extraction.
  • Bréard, Andrea. 1999. Re-Kreation eines mathematischen Konzepts im chinesischen Diskurs: ‘Reihen’ vom 1. bis zum 19. Jahrhundert. Stuttgart: Steiner. — Background on the recovery of Sòng-Yuán algebraic methods in the Qián-Jiā era.
  • Martzloff, Jean-Claude. 1997. A History of Chinese Mathematics. Berlin: Springer. — Treats Wáng Xiàotōng’s cubic methods and Lǐ Huáng’s kǎo-zhù exposition.
  • Qián Bǎocóng 錢寶琮, ed. 1963. Suàn-jīng shí-shū 算經十書. Beijing: Zhōnghuá shūjú. — Incorporates Lǐ Huáng’s emendations in the Jí-gǔ critical text.

Graph View

Backlinks