Cèliáng fǎyì 測量法義
The Meaning of the Measurement Methods by 徐光啟 (Xú Guāngqǐ, 1562–1633, 明, zhuàn 撰), translating the oral teachings of 利瑪竇 (Matteo Ricci)
About the work
A 1-juan trilogy (the catalog assigns single ID and single 卷-count, but the WYG actually bundles three independent works under the Cèliáng fǎyì heading): (I) Cèliáng fǎyì 測量法義 (1 juàn) — the principal work, an exposition of Western trigonometric measurement theory using the jǔdù 矩度 (graduated quadrant) instrument; (II) Cèliáng yìtóng 測量異同 (1 juàn) — a comparison of Western measurement methods with the Chinese jiǔzhāng gōugǔ 九章勾股 (right-triangle) tradition of the Jiǔzhāng suànshù 九章算術; (III) Gōugǔ yì 勾股義 (1 juàn) — an exposition of the underlying right-triangle theory. All three are by Xú Guāngqǐ, presenting in Chinese the mathematical-trigonometric methods orally taught him by Matteo Ricci. Composition began in dīngwèi (1607), immediately following the completion of the Ricci-Xú translation of the first six books of Euclid’s Elements (the Jǐhé yuánběn 幾何原本); the works are designed as practical applications of the new Euclidean geometry to surveying, height-and-distance measurement, and the foundational right-triangle relations. Xú’s preface explicitly states that Ricci had been “translating various measurement methods for ten years already” by 1607, but that “to attach meaning ( yì ) to the method ( fǎ )” — i.e. to give explanatory geometric proofs rather than mere computational recipes — became possible only after the completion of the Euclid translation. The trilogy is the foundational application of European Euclidean geometry to the Chinese surveying-and-measurement tradition, the second-generation work that turned the abstract Euclid into practical instrument-of-state.
Tiyao
[Sub-classification: 子部, Tiānwén suànfǎ class 1, tuībù sub-category. Edition: WYG.]
Respectfully examined: Cèliáng fǎyì 1 juàn, Cèliáng yìtóng 1 juàn, Gōugǔ yì 1 juàn — by Xú Guāngqǐ of the Míng. The first juàn extends what Lì Mǎdòu [Ricci] translated, in order to clarify the meaning of right-triangle measurement: first the construction of the instrument — the instrument is what the Zhōubì calls the jǔ (Try-square — graduated quadrant); next discusses shadows: shadows have upright (dǎo) and inverted (zhèng), namely what the Zhōubì calls the yǎngjǔ (upward-square), fùjǔ (downward-square), and wòjǔ (lying-square); next sets out fifteen propositions, in order to clarify the methods of measuring-and-observing height, depth, breadth, and distance — namely what the Zhōubì calls zhīgāo (knowing height), zhīyuǎn (knowing distance), and zhīshēn (knowing depth).
The next juàn takes the ancient method [as preserved in] the Jiǔzhāng (Nine Chapters on Mathematical Procedures)’ gōugǔ (right-triangle) measurements, and compares them with the new method to demonstrate their differences-and-similarities — in order to clarify that, although the ancient method of measurement is comprehensive, its underlying meaning is hidden. However, measurement is only one branch of gōugǔ; therefore in the third juàn he treats specifically the meaning of gōugǔ.
The preface cites the Zhōubì in order to clarify the source from which the establishment-of-method derives — and the rooting of the Western technique in this [Chinese tradition] is also implicitly visible.
His statement that Lǐ Yě 李冶 elaborated the right-triangle method in the Cèyuán hǎijìng 測圓海鏡 [does not show] understanding of the book’s purport; he further says he “wished to expound its meaning but had not had leisure” — but this means he had not [in fact] understood the lì tiānyuányī 立天元一 method, and was being a fancy-sayer. The ancient lì tiānyuányī method is the Western jiègēnfāng 借根方 method [algebra]. By this time the Westerners had been arriving for some years, and yet of [Lǐ] Yě’s book they still could not obtain its understanding. We can conclude that the jiègēnfāng method must have come out subsequently [to the Cèliáng trilogy].
The three juàn’s sequence is roughly thus, and their intent is in every case to illuminate the use of the Jǐhé yuánběn (Euclid’s Elements). For the ancient method rarely speaks of [a problem’s] meaning — when it does, it [merely] explains [each] problem [in turn]. Of the European learning, [it had] in earlier times [the figure] Ōujǐlǐdé 歐几里得 [Euclid], who according to the principles of triangles, squares, and circles propounded the various categories’ [geometric] reasoning, composing a book in 13 juàn called the Jǐhé yuánběn (later Lì Mǎdòu’s teacher [Christopher Clavius] Dīngshì added a continuation in 2 juàn, total 15 juàn). From this onward, all who studied calculation must necessarily first habituate themselves in his book. As for explaining the meaning of some method: when it is the same as something in the Jǐhé yuánběn, [the writer] only notes “see Jǐhé yuánběn, juàn such, section such”, and does not raise his words [again]; only what the Jǐhé yuánběn cannot reach does he [then] explain. This is the European-learning’s protocol.
Guāngqǐ, having translated with Lì Mǎdòu the first six juàn of the Jǐhé yuánběn, and wishing to use this book according to its protocol, therefore made this [trilogy] to set out the example.
His Cèliáng fǎyì preface says: “To establish meaning in the method (fǎ ér jì zhī yì yě) — this began from the year dīngwèi [1607]. Why was it deferred [until then]? At that time the six juàn of the Jǐhé yuánběn had only just been finished. Reaching this point, [the meaning] could be transmitted”. Hence one can know the intent of his composing the book.
Respectfully collated, Qiánlóng 46, twelfth month [January 1782].
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.
Original preface (Xú Guāngqǐ, 1607–1608)
Cèliáng fǎyì: Master XīTài’s [Ricci’s] translation of the various measurement methods has [now been ongoing for] ten years. To attach meaning (yì) to method (fǎ) — this began from the dīngwèi year [1607]. Why was it deferred [until then]? At that time the six juàn of the Jǐhé yuánběn had only just been finished. Reaching this [moment], one could thereafter transmit its meaning.
This method, with the Zhōubì-and-Jiǔzhāng’s gōugǔ cèwàng (right-triangle measurement-and-observation) — [is it] different or [is it] not? Not different. If not different, what is its value? Also valuable for its meaning. Liú Huī, Shěn Cúnzhōng [Shěn Kuò], and that current — all once spoke of measurement-and-observation, [but] could only explain a single gnomon, [they] could not explain double gnomons; spoke of the great-and-small gōugǔ being mutually-derivable on the [premise that] the small gǔ-greater gōu and the small gōu-greater gǔ, the two-contained-areas, are equal — but did not say why they must be equal in order to be mutually-derivable. Just like Master XīTài before [the year] dīngwèi. Why so? Without anything to serve as their foundation. Without anything to serve as their foundation — not only could the various gentlemen not say it, even Lì Shǒu [the legendary calendrical official] and Shāng Gāo could not say it. The Zhōubì does not speak of foundation? — not so: within its foundation there is yet another foundation. Without exhaustively explicating the Jǐhé yuánběn, one cannot stop. The Yuánběn’s capability for use is like this — and not yet exhausted. It is the xī [river-mouse] at the river, and the lí [conch] at the sea.
Why has [my work] taken these [three small problems] first? Because numbers are easy to see; small numbers are easy to explain. To broaden the technique and use it to govern water and govern fields — that benefit is great, that task is urgent. Therefore I take these first; subsequently there will be those who narrate, those who originate, who use it on a hundred-and-thousand-and-ten-thousand topics. As yet, [it is still] like drinking-from-the-river-with-a-spoon-from-the-sea — not yet exhausted. This is the meaning of the Yuánběn.
Abstract
Composition window: 1607 (Wànlì 35, dīngwèi — the year Xú Guāngqǐ states for the work’s beginning) – 1608 (the work was certainly complete by Xú’s appointment to Tianjin agricultural duty in 1608, after which his collaboration with Ricci took different forms). The trilogy is the immediate sequel to the Ricci-Xú translation of the first six books of Euclid’s Elements (Jǐhé yuánběn 幾何原本, completed bǐngwǔ 1606 and printed dīngwèi 1607).
The trilogy is foundational for two distinct reasons:
(a) Practical demonstration of Euclid’s utility: as Xú’s preface and the 提要 both emphasize, the works are designed as the demonstration of use (yòng 用) for the abstract geometric meaning (yì 義) just made available in the Jǐhé yuánběn. By taking the relatively elementary Chinese gōugǔ and cèwàng traditions and re-grounding them in Euclidean proof, Xú demonstrated that the new Western mathematics was not merely abstractly elegant but practically applicable to the surveying-and-measurement tasks that constituted the principal applied-mathematical work of late-imperial Chinese government. The 提要’s comment that “although the ancient method of measurement is comprehensive, its underlying meaning is hidden” is the standard late-imperial Chinese verdict on the difference between Chinese mathematical-procedural tradition and post-Euclidean European mathematical method: the procedures are comparable, but the axiomatic-deductive justification that distinguishes proof from recipe is novel.
(b) The jiègēnfāng / tiānyuányī identification: the 提要’s important historical-mathematical note — that the ancient Chinese lì tiānyuányī 立天元一 method (the polynomial-equation method developed by Lǐ Yě 李冶 in the 13th-century Cèyuán hǎijìng 測圓海鏡) is the same as the Western jiègēnfāng 借根方 method (algebra, transmitted to China later in the 17th century by Antoine Thomas, S.J., and integrated into the Kāngxī-period Shùlǐ jīngyùn) — is one of the foundational claims of the late-imperial “Western methods originate from the East” (Xīfǎ Zhōngyuán 西法中源) historiographical tradition. The 提要 is critical of Xú: had Xú properly understood Lǐ Yě’s tiānyuányī, he would have recognized that the Western Jesuits had nothing essentially new to teach in this domain. The criticism is mathematically dubious (post-Cardano-Bombelli European algebra was substantially more advanced than Lǐ Yě’s tiānyuányī in handling general polynomial equations) but historically revealing of late-Qīng anti-Western intellectual posture even within the broadly pro-Jesuit Sìkù synthesis.
The 提要’s note on Christopher Clavius’s continuation of Euclid (Dīngshì xù wéi èr juàn, gòng shíwǔ juàn “Master Dīng [Clavius] continued [Euclid] in two more juàn, total fifteen juàn”) is also notable: it reflects the Sìkù editors’ awareness that the Ricci-Xú translation went only as far as Book VI of Euclid (the canonical thirteen books plus Clavius’s two-book continuation = fifteen books) and stopped there. The “rest of Euclid” was not translated into Chinese until Alexander Wylie’s collaboration with Lǐ Shànlán 李善蘭 in the Xīnjiāng yīshū 新疆譯書 of 1857.
The methodological credo attributed to “the European-learning’s protocol” — that one need not re-explain a result already established in Euclid but only refer to it by juàn and jié — is a striking late-Qīng recognition of the Jǐhé yuánběn’s status as foundational reference work in the Western mathematical tradition. The 提要’s editor was effectively articulating the modern conception of axiomatic foundation as the Westernscientific-distinctive feature.
For Xú Guāngqǐ’s biography and broader intellectual program (Catholic conversion, agricultural reform, 1611-and-1629 calendar-reform memorials, the Chóngzhēn lìshū directorship), see 徐光啟. For the broader Jesuit-Chinese intellectual collaboration, see 利瑪竇.
Translations and research
- Engelfriet, Peter M. Euclid in China: The Genesis of the First Chinese Translation of Euclid’s Elements, Books I–VI (Jihe Yuanben, Beijing, 1607) and Its Reception up to 1723, Sinica Leidensia 40, Leiden: Brill, 1998. The standard study, with detailed treatment of the Cè-liáng trilogy as Xú’s sequel demonstration.
- Jami, Catherine. The Emperor’s New Mathematics: Western Learning and Imperial Authority during the Kangxi Reign (1662–1722), Oxford: Oxford University Press, 2012 (background on the longer reception of the Ricci-Xú geometric program).
- Hashimoto Keizō 橋本敬造. Hō Yū-ran: Christian Mission and Calendrical Reform in Late Ming China, Kyoto: Kansai University Press, 1988.
- Liáng Jiāmiǎn 梁家勉. Xú Guāngqǐ nián-pǔ 徐光啟年譜, Shànghǎi: Shànghǎi Gǔjí Chūbǎnshè, 1981.
- Sun, Xiaochun and Han Qi (eds.). History of Mathematics in China, in History of Mathematics in China (in Chinese, multiple volumes), Beijing: Science Press, various dates.
- Hé Bǐngyù (Ho Peng-Yoke). Li, Qi, and Shu: An Introduction to Science and Civilization in China, Hong Kong: Hong Kong University Press, 1985.
Other points of interest
The 提要’s Xīfǎ Zhōngyuán (Western methods originate from the East) move on the jiègēnfāng / tiānyuányī identification deserves comparison with the parallel move on the Zhōubì and the European astronomy in KR3f0001: both are characteristic Qián-lóng-period assertions of Chinese intellectual priority, framed as a nuanced acknowledgment that the Westerners brought refinements but not foundational novelty. The historical-mathematical accuracy of these claims varies considerably (the Zhōubì-Western correlation is largely conventional; the tiānyuányī-algebra correlation is partly genuine but substantially overstated).
Xú Guāngqǐ’s preface’s vivid metaphors — Euclid’s foundational character likened to the river-mouse drinking from the river and the conch from the sea (i.e. inexhaustible) — and his explicit identification of the social-utilitarian motive (“to govern water and govern fields”) for prioritizing surveying-applications, are characteristic of his pragmatic intellectual style and his self-conception as an applied mathematician working for the public good. The phrase “to govern water and govern fields” foreshadows his mature program (the Tàixī shuǐfǎ with de Ursis, 1612; the Nóng zhèng quán shū, posthumous 1639) of applied science in service of state-and-people.