Yuánróng jiàoyì 圜容較義
Comparative Meaning of [Areas Inscribed in] the Circle by 李之藻 (Lǐ Zhīzǎo, 1565–1630, 明, zhuàn 撰), translating teachings of 利瑪竇 (Matteo Ricci)
About the work
Lǐ Zhīzǎo’s 1-juan presentation of the European theorem of isoperimetric maximality — that among all plane figures with a given perimeter, the circle has the greatest area, and among all solids with a given surface area, the sphere has the greatest volume — together with the supporting Euclidean apparatus of inscribed-and-circumscribed regular polygons. The work was translated from Matteo Ricci’s oral exposition (which Lǐ Zhīzǎo expressly notes followed his earlier study of geometry under Xíng [Yúnlù?] gōng 邢公) and completed in ten days in 1608 (Wànlì wùshēn); first printed by Zhùshǐ Bì gōng 柱史畢公 (presumably Bì Gōngchén 畢拱辰) at Beijing, then re-engraved by Wāng Mèngpú 汪孟樸 in conjunction with his publication of the Ricci-Lǐ Zhīzǎo Tóngwén suànzhǐ in 1614 (Wànlì jiǎyín) — the date appearing in the Sìkù 提要 reference. The work consists of 5 definitions (wǔ jiè 五界) and 18 propositions (shíbā tí 十八題), presenting the theorem first for plane figures (the circle’s maximality among isoperimetric figures of any number of sides), then by analogy for solid figures (the sphere’s maximality among iso-area solids). The plane case is rigorously demonstrated; the solid case is presented analogically, by argument from the planar to the spherical (jiè píngmiàn yǐ tuī lìyuán, shè jiǎoxíng yǐ zhēng húntǐ 借平面以推立圜，設角形以徵渾體 — “borrowing the plane figure to derive the standing circle [sphere]; setting up the angle-figure to verify the sphere-body”).
The work is significant as the first Chinese exposition of an integrated geometric theorem (with proof) rather than a recipe or method; together with the Ricci-Xú Guāngqǐ Euclid translation KR3f0013 Jǐ-hé yuán-běn and the Xú trilogy KR3f0014 Cè-liáng fǎ-yì, it forms the second-generation foundation of post-Wànlì Chinese mathematical practice. Lǐ Zhīzǎo’s preface — one of the longest and most rhetorically elaborate of his prefaces — connects the geometric theorem to a broad cosmological-and-theological vision of the circle as the principle of natural form in everything from celestial bodies to seeds, eggs, vortices of water, and human anatomy (“the human skull, the eye-pupil, the ear-orifice — all are completed by being round”). The cosmological framing prepares the polemical critique that follows: a refutation of the Buddhist catuṣ-mahā-dvīpa cosmology (the Mount Sumeru with the four continents) on geometric grounds, and a rebuttal of the Liè-zǐ “two children disputing about the sun” episode (the question of whether the sun is closer at noon or at sunrise) by appealing to the European theory of atmospheric refraction.
Tiyao
[Sub-classification: 子部, Tiānwén suànfǎ class 1, tuībù sub-category. Edition: WYG.]
Respectfully examined: Yuánróng jiàoyì, 1 juàn, by Lǐ Zhīzǎo of the Míng — also what was transmitted by Lì Mǎdòu [Ricci]. In front [is] Zhīzǎo’s own preface from Wànlì jiǎyín [1614], which states:
“Of all things having form, only the round is greatest. Of all forms that may be received, only the round receives the most. The body of the round-sphere is hard to name, but the figure of the plane is easy to analyze. Try taking figures of the same circumference and comparing them: figures of equal sides necessarily exceed figures of unequal sides; figures with many sides necessarily exceed figures with few sides; the figure with the most sides is the circle; the figure with the most-equal sides is also the circle. To analyze [it]: its divisions and seconds [degrees] are not [merely] one-hundred-million [in number] — by which we know it has many sides. To connect [its sides]: there are no rough corners — by which we know its sides are equal. Where sides are equal but not many, [the figure] cannot become round; only with many sides and equal sides — therefore the round receives the greatest [area].
Formerly following Master Xíng [Yúnlù?] [I had been] investigating the celestial body. Hence in discussing the round-receiving [property] I extracted one principle, then arranged it as 5 definitions and 18 propositions, borrowing the plane figure to derive the standing circle [sphere], setting up the angle-figure to verify the sphere-body”, and so on.
For the figure has its full body, viewed as a single face — from one face one can extrapolate to the whole body. Therefore he says “borrowing the plane to measure the standing circle [sphere]“. A face must have boundaries; boundaries are lines and edges; two lines mutually supporting must form an angle. To analyze a round figure, [it] becomes [composed of] each [as] angle[-figure]; to combine angle[-figures], [they] together complete the round. Therefore he says “setting up angle[-figures] to verify the sphere-body”.
Although the book clarifies the meaning of round-receiving, the proportional meanings of each face and each body are everywhere apparent here; and in their successive generation [from one another] toward the Zhōubì’s [doctrine] that “the circle comes out of the square, and the square comes out of the carpenter’s-square” — these too are much illuminated.
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.
Abstract
Composition window: 1608 (Wànlì wùshēn, the year of completion stated in the preface — Lǐ Zhīzǎo notes the work was completed in ten days during a period of preparation for an official posting at Chánzhōu 澶州) – 1614 (Wànlì jiǎyín, the date of the printed preface and the publication of the Wāng Mèngpú reprint in conjunction with the Tóngwén suànzhǐ). The composition itself was rapid; the publication-and-recirculation involved two stages.
The mathematical content traces back to the standard scholastic-Euclidean treatment of the isoperimetric problem in the European medieval-Renaissance geometric tradition. The plane case (circle as the maximum of all isoperimetric polygons) had been fully demonstrated since the Greek antiquity (Pappus’s Synagoge book V); the solid analogue (the sphere as the maximum of all iso-area surfaces) was known but rigorously demonstrated only later. Ricci’s source for the Chinese transmission was almost certainly Christopher Clavius’s Euclidis Elementorum libri XV (Rome, 1574), with its expanded commentary, supplemented by other late-Renaissance European geometric texts. The Chinese rendering by Lǐ Zhīzǎo is faithful in mathematical content but expanded in cultural framing.
Two interpretive notes deserve attention:
(1) The work’s cultural framing of the circle is unusual in the Wànlì-period Jesuit corpus. Lǐ Zhīzǎo’s preface deploys the geometric theorem as the foundation of a broad cosmological-rationalist apologetic against Buddhist (Sumeru cosmology) and Daoist (Lièzǐ “two children disputing about the sun”) cosmographies. The polemical purpose is partly to vindicate the round-Earth and spherical-celestial-body doctrines that the Jesuits were transmitting; partly to claim for European-style demonstrative reasoning the cultural authority hitherto held by Buddhist-Daoist cosmological speculation. The 提要’s exclusion of all this polemical material from its summary — focusing exclusively on the mathematical content — reflects the Sìkù editorial preference for separating “Western technique” (preserved) from “Western polemic” (set aside, as in the KR3f0012 case).
(2) The 提要’s connection of the theorem to the Zhōubì’s “the circle comes out of the square” (圓出於方) doctrine is a characteristic Qián-lóng-period framing — assimilating the European result to the foundational Chinese text. Mathematically, the Zhōubì’s remark refers to the inscribed-square construction (the circle being inscribed in a circumscribing square through the gōugǔ relations); the Yuánróng jiàoyì’s isoperimetric theorem is logically distinct, but the editors’ juxtaposition is rhetorically effective. The same framing-strategy is at work in the KR3f0014 Cèliáng fǎyì 提要 and the KR3f0001 Zhōubì 提要.
Lǐ Zhīzǎo’s preface contains a passing reference to having “formerly followed Master Xíng to investigate the celestial body” (xī cóng Xínggōng yánqióng tiāntǐ 昔從邢公研窮天體). The “Master Xíng” is conventionally identified as Xíng Yúnlù 邢雲路 (KR3f0008 / 邢雲路), the leading independent calendrical critic of the late-Wànlì period. If correct, this identification establishes a direct intellectual filiation from Xíng Yúnlù’s late-Wànlì calendar-reform agitation to Lǐ Zhīzǎo’s Jesuit-collaborative mathematical work — placing both within a unified late-Wànlì astronomical-reform circle. (Some scholars have suggested that the “Master Xíng” might instead be Xíng Tóng 邢侗 the calligrapher, but this seems less likely given the context “to investigate the celestial body”.)
For Lǐ Zhīzǎo’s broader career, see 李之藻. For the Jesuit-Chinese collaborative mathematical-astronomical project, see 利瑪竇, 徐光啟, KR3f0009, KR3f0014, KR3f0015.
Translations and research
- Engelfriet, Peter M. Euclid in China: The Genesis of the First Chinese Translation of Euclid’s Elements, Sinica Leidensia 40, Leiden: Brill, 1998 (treats the Lǐ Zhīzǎo-Ricci collaboration in context).
- Hashimoto Keizō 橋本敬造. Hō Yū-ran: Christian Mission and Calendrical Reform in Late Ming China, Kyoto: Kansai University Press, 1988.
- Standaert, Nicolas (ed.). Handbook of Christianity in China, vol. 1, Leiden: Brill, 2001.
- Hé Bǐngyù (Ho Peng-Yoke). Li, Qi, and Shu: An Introduction to Science and Civilization in China, Hong Kong: Hong Kong University Press, 1985.
- Mei Rongzhao 梅榮照, Míng-Qīng shù-xué-shǐ lùn-wén jí 明清數學史論文集, Nánjīng: Jiāngsū Jiào-yù Chūbǎnshè, 1990 (treats Lǐ Zhīzǎo’s mathematical contributions).
Other points of interest
The 提要’s preservation of Lǐ Zhīzǎo’s preface intact is notable: the 提要 of the related work KR3f0012 Tiānwèn lüè deleted the original preface for theological content, but the Yuánróng jiàoyì preface was preserved despite its cosmological-polemical content. The difference may reflect the Sìkù editors’ judgment that Lǐ Zhīzǎo’s polemic, being directed against Buddhist and Daoist cosmologies (rather than promoting Catholic theology), was within the bounds of acceptable late-Confucian critical discourse.
The work’s title — Yuánróng 圜容 (“circle-receiving”) — is a technical translation of the Greek-derived European isoperimetric concept. The term was Lǐ Zhīzǎo’s coinage and would remain the standard Chinese mathematical translation of “isoperimetric” through the late Qīng.