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海島算經細草圖說

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Text ID
KR3fc003
Date
1800–1820
Dynasty
清

Hǎidǎo suànjīng xìcǎo túshuō 海島算經細草圖說

Detailed Procedure and Diagrammatic Exposition of the Sea Island Mathematical Canon by 李潢 (撰), 沈欽裴 (校訂)

About the work

The companion piece to Lǐ Huáng’s KR3fc002 Jiǔzhāng suànshù xìcǎo túshuō, applying the same detailed-procedure-and-diagrammatic method to 劉徽 Liú Huī’s Hǎidǎo suànjīng 海島算經 (KR3f0035) — the nine-problem treatise on right-triangle remote-sensing techniques (originally an appendix-chapter of Liú Huī’s Jiǔzhāng commentary under the title Chóngchā 重差, separately canonized by the Táng). In 2 juàn. Compiled in the closing decade of Lǐ Huáng’s life and put through final editing and publication by 沈欽裴 Shěn Qīnpéi after Lǐ’s death in 1811.

Abstract

The Hǎidǎo suànjīng survives only as a fragmentary 9-problem text recovered from the Yǒnglè dàdiǎn 永樂大典 by Dài Zhèn 戴震 in the Sìkù compilation; Liú Huī’s original Chóngchā chapter (mentioned in his preface to the Jiǔzhāng commentary as containing the techniques for surveying inaccessible distances by right-triangle sighting) is otherwise lost. Lǐ Huáng’s task in the present work was thus reconstructive: given only the problem-statements and the bald numerical answers preserved in the recovered text, recover (a) the implicit procedural calculation by which each answer was reached, and (b) the geometric demonstration justifying the procedure. Each of the nine problems — the sea-island whose height must be determined by twin sightings from poles, the height of a hill seen at a distance, the depth of a ravine, the height of a pagoda, the size of a river, the area of a city seen from afar, and so on — is presented with full xìcǎo working and túshuō figure-and-explanation reconstructing the surveying geometry.

The work is the principal scholarly exposition of the Hǎidǎo in late-imperial Chinese mathematical literature; its diagrammatic reconstructions of the chóngchā (double-difference) sighting method are still followed in modern textbook accounts. Lǐ Huáng’s analysis showed that all nine problems reduce to applications of essentially three procedures: (i) the chóngbiǎo 重表 (double-pole) procedure for inaccessible elevations; (ii) the chóngchā 重差 (double-difference) procedure proper, requiring two sightings from a known baseline; and (iii) the liánsuǒ 連索 (linked-sighting) procedure for problems requiring more than two sightings. This systematization shaped 20th-century scholarly understanding of the Hǎidǎo — for example Frank Swetz’s The Sea Island Mathematical Manual (1992) follows Lǐ Huáng’s procedural reconstructions throughout.

Lǐ Huáng died (1811, CBDB id 59341) before the work reached publication; 沈欽裴 Shěn Qīnpéi put it through final editing and brought it to print, together with the Jiǔzhāng and the KR3fc005 Jígǔ suànjīng kǎozhù, around 1820. The trio constitute the most systematic Qīng-period scholarly recovery of the procedural specifics of the Suànjīng shíshū 算經十書 mathematical curriculum.

Translations and research

  • Swetz, Frank J. 1992. The Sea Island Mathematical Manual: Surveying and Mathematics in Ancient China. University Park: Pennsylvania State University Press. — The standard English-language study of the Hǎi-dǎo; follows Lǐ Huáng’s procedural reconstructions.
  • Martzloff, Jean-Claude. 1997. A History of Chinese Mathematics. Berlin: Springer. — Treats Lǐ’s Hǎi-dǎo exposition in the context of Qián-Jiā mathematical philology.
  • Guo Shuchun 郭書春. 2009. Hǎi-dǎo suàn-jīng yì-jiě 海島算經譯解. Modern annotated translation that incorporates Lǐ Huáng’s procedural reconstructions.

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