Zhōubì suànjīng 周髀算經
The Mathematical Classic of the Zhōu Gnomon by 趙君卿 (Zhào Jūnqīng / Zhào Shuǎng, 漢-Wèi, commentary); 甄鸞 (Zhēn Luán, 北周) — re-elaboration; 李淳風 (Lǐ Chúnfēng, 602–670, 唐) — imperial annotation; 李籍 (Lǐ Jí, 唐) — phonetic-and-meaning gloss (1 appended juan)
About the work
The foundational Chinese mathematical-astronomical classic, in 2 juan + 1 juan of phonetic gloss, embodying the Gàitiān cosmology (蓋天 “Heaven-as-Cover”, the older Chinese astronomical model in which Heaven is a circular dome covering a flat disk-Earth). The work’s first chapter — the famous dialogue between the Duke of Zhōu (Zhōu Gōng) and Shāng Gāo 商高 — is the locus classicus of the Pythagorean / right-angle (gōugǔ 勾股) theorem in Chinese mathematics, predating Pythagoras-derived Greek formulations. Subsequent material develops: the use of an 8-foot vertical gnomon (bì 髀 = gnomon-foot) at the Zhōu locale to measure the sun’s solstitial-and-equinoctial shadow; the seven-concentric-circle (qīhéng liùjiān 七衡六間) model of the sun’s annual circuit at the celestial pole; the xuánjī (璇璣) circumpolar-spinning system; and other mathematical-astronomical doctrines. Composition is conventionally placed between the late Western Hàn (-100) and the late Eastern Hàn (+200), but the work’s mathematical content includes pre-Hàn material going back to the Warring States. 趙君卿 Zhào Jūnqīng’s late-Hàn / WèiJìn commentary supplies the proof of the Pythagorean theorem via red-and-yellow shaded-area diagrams (zhūhuáng zhī shí) — the earliest such proof in Chinese mathematics. The work was commissioned annotated by 李淳風 Lǐ Chúnfēng under the Táng imperial Shí bù suàn jīng 十部算經 (Ten Mathematical Classics) project.
Tiyao
[Sub-classification: 子部十六 天文算法類一. Edition: 永樂大典本.]
According to the Suí shū jīngjí zhì: in the Astronomy class, first listed is Zhōubì 1 juan, with Zhào Yīng’s commentary; another 1 juan with Zhēn Luán’s re-elaboration. The Táng shū yìwén zhì lists Lǐ Chúnfēng’s Shì Zhōubì in 2 juan, alongside Zhào Yīng and Zhēn Luán’s commentaries, in the Astronomy class; and in the Calendrical-Mathematics class, a Lǐ Chúnfēng zhù Zhōubì suànjīng in 2 juan — repeated entry of one book.
The book’s content: “The Zhōubì, 8 feet long; on the summer-solstice day, the gnomon-shadow is 1 chǐ 6 cùn” — bì 髀 is gǔ 股 (the [right-angle] vertical leg). At the Zhōu locality establishing an 8-foot gnomon as the gǔ; its shadow is the gōu 勾 (horizontal leg) — therefore “Zhōubì” (Zhōu-gnomon).
The first chapter — Zhōu Gōng’s question-and-answer with Shāng Gāo — is in fact the ancestor-archive (鼻祖) of the gōugǔ (right-angle / Pythagorean) theorem. Therefore the imperially-composed Shùlǐ jīngyùn 數理精蘊 records it at the head and explains it in detail, calling it “the inheritance of the Six-Arts canon of the ChéngZhōu period”.
The “Ròng Fāng’s question to Chén Zǐ” passage and following — Xú Guāngqǐ called this “the great folly of all time”. Today examining the text carefully: it discusses only north-south shadow-difference, taking the earth as flat-and-far, and again with flat-and-far measuring the heavens — truly a fanciful argument. But this is utterly different in style from the main text, suggesting later tradition that has been mistakenly admitted as main-text — like the Xià xiǎozhèng’s mixed-classic-and-commentary, before Fù Sōngqīng’s collation, no one could read [it].
The main text’s broad-and-precise content, however, all preserves the meaning of ancient method and opens the source of Western method. For example, in the book the xuánjī (璇璣) in one day-and-night encircles the North-Pole once and over by one dù (degree), terminating at midnight at the xuánjī’s starting position below the Pole at the zǐ (子) position; at spring-equinox-midnight starting at the Pole’s left mǎo (卯) position; at summer-solstice-midnight starting at the Pole’s upper wǔ (午) position; at autumn-equinox-midnight starting at the Pole’s right yǒu (酉) position — this is the xuánjī’s westward-circuit-extreme, never-changing-since-antiquity. The seven-concentric-circle six-interval (qīhéng liùjiān) measurement of the sun’s fāliǎn (orbital extreme): winter-solstice the sun is at the outer ring; summer-solstice the sun is at the inner ring; spring-equinox-and-autumn-equinox at the middle ring; at-the-ring-corresponds-to-major-qì-period; in-the-interval-corresponds-to-minor-qì-period — also never-changing-since-antiquity. This is the ancient Gàitiān (Cover-Heaven) doctrine’s surviving method.
The Húntiān (Sphere-Heaven) is like a globe, with the star-images written on the outside; people from outside the heavens look at the heavens. The Gàitiān is like a hat, with the star-images written on the inside; people from inside the heavens look at the heavens. The hat-shape is half-round, like a stretched cover — therefore called Gàitiān. Combining the upper and lower halves of the earth-sphere is the heavens’ round-sphere body.
The Gàitiān doctrine has been lost for a long time. Therefore from the Hàn down through Yuán-and-Míng, all have followed the Húntiān. Under the Wànlì period of the Míng, the Europeans entered China and first established a new method, claiming it precise. But their statement that the earth is round — exactly the Zhōubì’s “the earth’s law is the inverted-bowl, with the four sides falling-away” (地法覆槃，滂沱四隤而下); their statement of north-south lǐchā (mileage-difference) — exactly the Zhōubì’s “at the Pole’s left-and-right, summer has unmelted ice; at the middle-ring’s left-and-right, winter has unwithering grass; the five grains [in the south] one year ripen twice” — this is hot-and-cold variation depending on north-south difference. Their statement that “from spring-equinox to autumn-equinox, [under] the Pole there is always sunlight; from autumn-equinox to spring-equinox, [under] the Pole there is always no sunlight” — this is day-and-night length variation depending on north-south difference.
Their statement of east-west lǐchā (mileage-difference) — exactly the Zhōubì’s “east-side noon, west-side midnight; west-side noon, east-side midnight; day-and-night exchanging places, like four-seasons mutually-opposite” — this is solar-term combining-with-new-moon timing variation depending on east-west difference.
Further, Lǐ Zhīzǎo 李之藻 used Western methods to make the Húngài tōngxiàn 渾蓋通憲, expanding the short-day ring to be larger than the equator-ring — exactly the Zhōubì’s outer-ring expansion to be larger than the middle-ring. The new-method calendar-book, transmitting Tycho [Brahe]‘s pre-Western method’s 365 1/4-day, every 4 years’ minor surplus making 1 day — exactly the Zhōubì’s “365 days in three [years] and 366 days in one”. Western methods derive from the Zhōubì — these are clear proofs. Only the later measurements have refined and increased — increasingly precise by progression.
The Míng shǐ Lì zhì says: “In Yáo’s time, [the imperial astronomers] dwelt in the West at Mèigǔ; the descendants of the calendrical-officers scattered into distant regions, which is why they were transmitted as Western learning.” This has its grounding.
This book’s printed copies have many errors and unreadable passages. We have now collated against the Yǒnglè dàdiǎn version, supplementing 147 missing characters, correcting 113 erroneous-and-disordered, deleting 18 redundant-and-duplicate. The old base recension is titled Hàn Zhào Jūnqīng zhù; his own preface says “Shuǎng 爽 due to my obscure-shielding…”; in the commentary he repeatedly says “Shuǎng 爽 perhaps doubts”, “Shuǎng 爽 not yet heard before” — so Shuǎng is Jūnqīng’s personal name. So the Suí and Táng zhì’s “Zhào Yīng” is presumably a graphic-corruption-error for “Zhào Shuǎng”. The commentary cites the Língxiàn and Qiánxiàng — so the man is after Zhāng Héng and Liú Hóng.
The old [recension] has Lǐ Jí’s yīnyì (phonetic-and-meaning gloss) — separately one juan; we retain it as before. Within the book, of the 5 figures, 3 have been lost and 1 is corrupt. We have respectfully supplemented based on the main-text and commentary.
Of the ancient jiǔ shù (nine mathematical methods), only the Jiǔzhāng and Zhōubì — these two books — are transmitted from earliest antiquity, and corrupt-errors are also particularly heavy. But by tracing the source-and-stream and obtaining the threads-and-strands, [the Zhōubì] is truly the great-treasure of the shùshù-jiā (numerical-arts school).
(Respectfully verified.)
Abstract
Composition window: 100 BCE – 200 CE. The Zhōubì’s base text is conventionally dated to the late Western Hàn (when the imperial astronomical bureaus standardized older Warring-States and early-Hàn material into canonical form), with extensive incorporation of pre-Hàn (Warring-States, possibly Spring-and-Autumn) mathematical content including the famous Zhōu Gōng / Shāng Gāo dialogue. The Zhào Jūnqīng commentary belongs to the late Hàn or WèiJìn period (ca. 200–300 CE).
The work’s significance:
(a) The foundational Chinese mathematical classic: alongside the Jiǔzhāng suànshù 九章算術 (KR3f0011?), the Zhōubì is one of the two oldest surviving Chinese mathematical works. Through the Zhōubì, the Chinese tradition’s distinct mathematical-and-astronomical heritage — gōugǔ (right-angle) theorem, qīhéng liùjiān astronomy, Gàitiān cosmology — entered the canonical record.
(b) The earliest Chinese proof of the Pythagorean theorem: the Zhào Jūnqīng commentary’s zhūhuáng zhī shí (red-and-yellow shaded-area) proof of the gōugǔ theorem — using bèichā jiǎnbìng (multiply-subtract-add-combine) operations on areas — is the earliest known Chinese proof and one of the earliest geometric proofs anywhere.
(c) The Gàitiān cosmology: the Zhōubì’s “Heaven as Cover” cosmology — the older Chinese astronomical model preceding the Hàn-period Húntiān (Sphere-Heaven) revolution — is one of the principal sources for the history of Chinese astronomical thinking.
(d) The “Western methods derive from the Zhōubì” thesis of the SKQS editors: a remarkable late-Qīng / early-modern attempt to reconcile the foreign provenance of Western post-Jesuit astronomical methods with classical Chinese authority by claiming pre-Western priority. The thesis is methodologically interesting both as scientific history (some of the parallels are real) and as cultural-political response to Western astronomical influence (the editors’ approval of Lǐ Zhīzǎo’s syncretic instruments and the Míng shǐ’s “Yáo’s astronomical officers scattered to the West” suggestion).
(e) The Yǒnglè dàdiǎn recovery’s textual restoration: 147 characters supplemented, 113 corrected, 18 deletions — one of the better-documented mid-Qīng textual-philological recoveries.
The catalog meta dynasty 漢 is consistent with the broad Hàn-period dating; lifedates-and-attribution information is in the 趙君卿, 甄鸞, 李淳風, 李籍 person notes.
Translations and research
- Cullen, Christopher. Astronomy and Mathematics in Ancient China: The Zhou Bi Suan Jing, Cambridge: Cambridge University Press, 1996. The standard scholarly English translation, with extensive philological apparatus.
- Joseph Needham (with Wáng Líng), Science and Civilisation in China, vol. 3 (Mathematics and the Sciences of the Heavens and the Earth), Cambridge: Cambridge University Press, 1959 (extensive treatment of the Zhōubì).
- Mǎ Bóyīng 馬伯英, Zhōngguó shù-xué shǐ 中國數學史, Tài-běi: Jiǔ-zhōu Tú-shū, 2010.
- Qián Bǎo-cóng 錢寶琮 (ed.), Suàn-jīng shí-shū 算經十書, 2 vols., Beijing: Zhōnghuá Shū-jú, 1963. The standard modern critical edition of the Ten Mathematical Classics including the Zhōubì.
Other points of interest
The Zhōubì’s status as one of the oldest surviving Chinese mathematical works — likely older than its companion the Jiǔzhāng suànshù — gives it special importance in the history of Chinese science. The Zhōu Gōng / Shāng Gāo dialogue’s articulation of the gōugǔ relationship “the gōu is 3, the gǔ is 4, the diagonal is 5” is one of the canonical statements of pre-modern Chinese mathematical reasoning.
The “Western methods derive from the Zhōubì” thesis articulated by the SKQS editors is a key text in the history of Chinese-Western scientific encounter. Modern history-of-science scholarship has rejected the literal claim of Western derivation from the Zhōubì but has shown that the parallels noted by the SKQS editors are real — independent ancient cosmological-and-mathematical traditions converged on similar conclusions about the spherical earth, north-south solar variation, and time-zone differences.