Hǎidǎo suànjīng 海島算經
Sea-Island Mathematical Classic by 劉徽 (Liú Huī, 魏-Jìn, zhuàn 撰); imperially commissioned annotation by 李淳風 (Lǐ Chúnfēng, 唐, fèngzhào zhù 奉詔注)
About the work
A 1-juan compendium of nine remote-sensing problems by Liú Huī, originally composed as the Chóngchā 重差 (Double-Differences) supplementary chapter appended to his commentary on the Jiǔzhāng suànshù (KR3f0032), separately catalogued and re-titled Hǎidǎo by the Tang Imperial Academy (656 CE) when the work was incorporated into the Suànjīng shíshū curriculum. The new title comes from the work’s opening problem: a sea-island whose distance and height must be determined from two paired sighting-stations on the mainland, using the chóngchā (double-difference) method that Liú Huī had developed as the foundational technique of pre-trigonometric Chinese remote-sensing computation.
The 9 problems address increasingly complex remote-sensing scenarios: (1) the height-and-distance of a sea-island; (2) the height of a tree on a hill; (3) the area of a square city; (4) the depth of a chasm; (5) the height of a building seen from below; (6) the width-and-depth of a river-valley; (7) the depth of a pool seen from above; (8) the breadth of a river crossed by a ferry; (9) the size and distance of a city seen from a distant elevated viewpoint. Each problem demonstrates the application of the chóngchā method by paired sighting-and-shadow measurements taken from two stations of known separation. The technique is the pre-trigonometric Chinese counterpart of European Renaissance triangulation methodology.
The Sui Jīngjí zhì lists the work as part of an enlarged 10-juàn Jiǔzhāng recension by Liú Huī (i.e., the original Jiǔzhāng nine juàn + Liú Huī’s Chóngchā as the 10th); the same Sui catalog separately lists a Liú Huī Jiǔzhāng Chóngchā tú in 1 juàn (the same Chóngchā material in independent circulation). The Tang Yìwén zhì lists the work as a separately-circulated 1-juàn text under the title Hǎidǎo. The Sìkù-recension is recovered from the Yǒnglè dàdiǎn.
Tiyao
Respectfully examined: Hǎidǎo suànjīng, 1 juàn, by Liú Huī of the Jìn, by-imperial-decree annotated by Lǐ Chúnfēng et al. of the Tang.
According to Liú Huī’s preface to the Jiǔzhāng suànshù: “[I,] Huī, sought into the Nine Numbers and there was the name chóngchā (double-differences). Whenever in observing the highest, measuring the deepest, and at the same time knowing the most distant — one must use chóngchā*. [I] then composed the* chóngchā and made annotations to it, in order to pursue the ancients’ purport, attached below the gōugǔ [chapter]. For measuring height, double the gnomon; for measuring depth, accumulate the carpenter’s-square; for the isolated, three-sights; for the isolated and laterally-sought, four-sights”. By this — Huī’s book is originally named Chóngchā, originally [had] no Hǎidǎo heading; also only attached below the gōugǔ [chapter] — not separately a book.
Therefore the Suí zhì’s Jiǔzhāng suànshù increased to 10 juàn, with “Liú Huī zhuàn” below — indeed taking the Jiǔzhāng nine juàn combined with this to make 10. But the Suí zhì and Táng zhì further both have Liú Huī Jiǔzhāng chóngchā tú 1 juàn — indeed that book also independently circulates, therefore separately recorded — one book appearing twice. Reaching the Táng zhì combined-listing of Liú Xiàng Jiǔzhāng chóngchā 1 juàn — then [Liú] Huī’s Chóngchā having already become its own juàn, accordingly Liú Huī was wrongly transcribed to Liú Xiàng — and one book appears three times.
Now in detail [we] examine: [we] fix [it] as Liú Huī’s book. As to the Hǎidǎo name, although in antiquity not seen, [it is] no more than that later people, on account of the volume’s head [problem of] Hǎidǎo [where the gnomon] is set up to ask [a] question, accordingly altered this name. However, the Táng Xuǎnjǔ zhì calls “the algebra-students [study] the Jiǔzhāng and Hǎidǎo together with a 3-year period [of study], [the test] questioning 3 items of Jiǔzhāng and 1 item of Hǎidǎo*” — then the alteration of title to Hǎidǎo from the early Táng was already so.
The book in the world has no transmitted recension; only it is dispersed-and-seen in the Yǒnglè dàdiǎn. Now [we have] gathered-and-edited it, sequenced still as 1 juàn. The fascicle [is] not large but the ancient method is fully present; therefore [it] should be with the Jiǔzhāng suànshù together expounded, in order to display the source from which the mathematical-arts family [traces its] origin.
Respectfully collated, Qiánlóng 46, second month [March 1781].
Chief Compilers: (subject) Jì Yún 紀昀, (subject) Lù Xíxióng 陸錫熊, (subject) Sūn Shìyì 孫士毅. Chief Collator: (subject) Lù Fèichí 陸費墀.
Abstract
Composition: c. 263–280 CE — coinciding with Liú Huī’s commentarial work on the Jiǔzhāng (begun 263 CE in the Wèi Jǐngyuán 4) and continuing into the Western Jìn period after 265. The Hǎidǎo / Chóngchā material was originally an integrated supplementary chapter of the Jiǔzhāng commentary, only later (in independent transmission and Tang re-titling) becoming a separately-catalogued work.
The work’s significance:
(a) Foundation of pre-trigonometric remote sensing: the chóngchā (double-difference) method developed in the work is the foundational technique of pre-trigonometric Chinese remote sensing, comparable to European Renaissance triangulation. Although less general than Hellenistic-Islamic spherical trigonometry, the chóngchā method is sufficient for a wide range of practical surveying-and-astronomical applications.
(b) Extension of the Jiǔzhāng gōugǔ tradition: the work extends the Jiǔzhāng’s foundational right-triangle (gōugǔ) chapter into the more elaborated domain of paired-sighting computation. The theoretical machinery of the gōugǔ is elementary; the Hǎidǎo’s 9 problems are sophisticated applications of that elementary machinery.
(c) Tang-period institutional canonization: the Hǎidǎo was the second principal text (after the Jiǔzhāng) of the Tang Imperial Academy Suànxué guǎn curriculum — the formal mathematical pedagogy that would shape East-Asian mathematical education for centuries.
For the foundational Jiǔzhāng, see KR3f0032. For the principal author, see 劉徽. For the Tang annotator, see 李淳風.
Translations and research
- van Hée, Louis. Le Classique de l’Île Maritime: Ouvrage chinois du IIIe siècle, in Quellen und Studien zur Geschichte der Mathematik, Astronomie und Physik, Berlin: Springer, 1932. The standard French translation.
- Swetz, Frank J. The Sea Island Mathematical Manual: Surveying and Mathematics in Ancient China, University Park: Pennsylvania State University Press, 1992. The standard English translation.
- Martzloff, Jean-Claude. A History of Chinese Mathematics, Berlin: Springer, 1997.
- Needham, Joseph (with Wang Ling), Science and Civilisation in China, vol. 3.
Other points of interest
The 提要’s reconstruction of the work’s titular history — from Liú Huī’s original Chóngchā appendix to the Sui-period Jiǔzhāng 10-juàn recension to the Tang-period independent Hǎidǎo 1-juàn — is one of the more thorough exercises in textual-genealogical reconstruction in the Tiānwén suànfǎ 提要 corpus. The identification of the Liú Huī → Liú Xiàng graphic-corruption as the cause of the Yìwén zhì’s separate listing is a refined textual-critical observation.