Xīnfǎ suànshū 新法算書
Computational Treatises of the New Method (originally Chóngzhēn lìshū 崇禎曆書, Calendar of the Chóngzhēn Era) by 徐光啟 (Xú Guāngqǐ, 1562–1633, 明, zhuàn 撰 — chief director 1629–1633); 李之藻 (Lǐ Zhīzǎo, 1565–1630, 明, zhuàn — co-director 1629–1630); 李天經 (Lǐ Tiānjīng, 1579–1659, 明, zhuàn — succeeding director 1633–1644); with the Western collaborators 龍華民 (Niccolò Longobardo, S.J., 1559–1654), 鄧玉函 (Johann Schreck-Terrentius, S.J., 1576–1630), 羅雅谷 (Giacomo Rho, S.J., 1593–1638), and 湯若望 (Adam Schall von Bell, S.J., 1591–1666)
About the work
The Chóngzhēn lìshū — re-titled Xīnfǎ suànshū under Qīng — is the great late-Míng calendrical-and-mathematical-astronomical compendium, compiled 1629–1635 under imperial commission as the foundational reform of the inherited Yuán-derived Dàtǒng lì. In 100 juàn (the Sìkù recension), the work synthesized the entire European mathematical-astronomical curriculum then accessible to the Beijing Jesuit residence — Tycho Brahe’s geo-heliocentric system as elaborated in the Astronomiae instauratae progymnasmata (1602), Kepler’s elliptical-orbit reform as transmitted by Schreck-Terrentius in the Tabulae Rudolphinae (1627), and the broader scholastic-astronomical framework of post-Clavius Roman College mathematics — with the Chinese state-astronomical tradition descending from the Yuán Shòushí lì 授時曆. The work was promulgated by the Qīng (1645) as the foundation of the Shíxiàn lì 時憲曆, the official imperial calendar from then until 1911 — making the Xīnfǎ suànshū arguably the single most consequential scientific work of late-imperial Chinese history.
The 100-juàn arrangement is canonical. The 提要 specifies eleven sections (shíyī bù 十一部) in two divisions:
Foundational Five (基本五目 jīběn wǔmù):
- Fǎyuán 法原 (Method-Origins — the foundational physical-and-mathematical principles)
- Fǎshù 法數 (Method-Numbers — numerical tables)
- Fǎsuàn 法算 (Method-Computation — computational procedures and formulas)
- Fǎqì 法器 (Method-Instruments — astronomical instruments)
- Huìtōng 會通 (Comprehensive-Synthesis — chronological and unit-conversion concordances between European and Chinese systems)
Sequential Six (節次六目 jiécì liùmù):
- Rìchán 日躔 (Sun-Position)
- Héngxīng 恒星 (Fixed Stars — including the first star catalog of 1812 stars by ecliptic coordinates ever produced for the Chinese audience)
- Yuèlí 月離 (Moon-Path)
- Rìyuè jiāohuì 日月交會 (Solar-Lunar Conjunctions and Eclipses)
- Wǔwěixīng 五緯星 (Five Planets)
- Wǔxīng jiāohuì 五星交會 (Five-Planet Conjunctions)
The work opens with Xiūlì yuánqǐ 修厯緣起 (Background of the Calendar-Reform — including the various memorials, the contemporary anti-reform polemics by Wèi Wénkuí 魏文魁, and Xú Guāngqǐ’s response Xuélì xiǎobiàn 學厯小辨), and closes with two appendices added by Schall von Bell in the early Qīng: the Lìfǎ Xīchuán 厯法西傳 (Western Transmission of the Calendrical Methods — Schall’s narrative of the project’s European mathematical-astronomical sources) and the Xīnfǎ biǎoyì 新法表異 (Differences-Tabulated of the New Method — comparative table demonstrating the new system’s superior accuracy against the Dàtǒng and other inherited methods).
Tiyao
[Sub-classification: 子部, Tiānwén suànfǎ class 1, tuībù sub-category. Edition: WYG.]
Respectfully examined: Xīnfǎ suànshū, 100 juàn — the Western-style new calendar revised by Grand Secretary Xú Guāngqǐ, Tàipúsì shàoqīng Lǐ Zhīzǎo, and Guānglùsì qīng Lǐ Tiānjīng of the Míng, together with the Western men Lóng Huámín, Dèng Yùhán, Luó Yǎgǔ, Tāng Ruòwàng, and others.
After Chénghuà of the Míng, the calendrical method became increasingly erroneous, yet the Bureau Officials clung blindly to the inherited tradition, and the court likewise feared change; those who proposed reforms were uniformly blocked and not put into action. In the Wànlì period, the great-Western-Sea men Lóng Huámín and Dèng Yùhán arrived successively at the capital — both deeply versed in calendrical method. Wǔguānzhèng Zhōu Zǐyú 周子愚 周子愚 requested that they be ordered to consult-and-revise [the existing methods]; the Ministry of Rites accordingly recommended Guāngqǐ and Zhīzǎo to take charge of the matter, but ordinary affairs interfered and there was no leisure to open the bureau.
Reaching Chóngzhēn 2 (1629), when the prediction of a solar eclipse failed of verification, the Ministry of Rites then for the first time memorialized to open a bureau and revise [the calendar], with Guāngqǐ in charge.
At that time the Mǎnchéng commoner Wèi Wénkuí 魏文魁 had composed two books, Lìyuán 厯元 and Lìcè 歷測, and had his son submit them to court. Guāngqǐ composed the Xuélì xiǎobiàn 學厯小辨 to refute their errors, and Wénkuí’s argument was accordingly dismissed.
On this basis, Guāngqǐ supervised the completion of dozens of juàn of the calendar-book, presenting them in succession. Then Guāngqǐ fell ill and died; Lǐ Tiānjīng succeeded him in directing the matter, and continued to present the calendar-book and instruments composed [thereafter]. The book in total has eleven sections:
[Foundational five (basic categories):] Fǎyuán (Method-Origins), Fǎshù (Method-Numbers), Fǎsuàn (Method-Computation), Fǎqì (Method-Instruments), Huìtōng (Comprehensive-Synthesis) — these are called the “Foundational Five Categories” (jīběn wǔmù).
[Sequential six (procedural categories):] Rìchán (Sun-Position), Héngxīng (Fixed Stars), Yuèlí (Moon-Path), Rìyuè jiāohuì (Solar-Lunar Conjunctions), Wǔwěixīng (Five Planets), Wǔxīng jiāohuì (Five-Planet Conjunctions) — these are called the “Sequential Six Categories” (jiécì liùmù).
The book’s beginning is the Xiūlì yuánqǐ 修厯緣起 (Background of the Calendar-Reform) — all the contemporary memorials and the matters of investigation-and-debate. The book’s end has the two kinds Lìfǎ Xīchuán 厯法西傳 (Western Transmission of the Calendrical Methods) and Xīnfǎ biǎoyì 新法表異 (Differences-Tabulated of the New Method) — these were composed by Tāng Ruòwàng [Adam Schall] after his entry into Our Dynasty [the Qīng], and were appended-and-printed for circulation.
Within [the work] there are explanations, there are techniques, there are diagrams, there are investigations, there are tables, there are discussions — all probing the depths and seeking the hidden, closely conforming to the heavens’ motion, sufficient to exhaust the treasury of European calendrical learning.
However, at that time [the work was] entangled in the court ministers’ factionalism. Although a decree was issued establishing two bureaus [for testing the new and old methods in parallel], and although year-after-year of measurements clearly demonstrated the new method’s precision, in the end [the new method] could not be put into operation.
Reaching the Sage Era’s [the Qīng’s] dragon-rising, [the new method’s] completed volumes were thereupon used to provide for the calendrical specialists’ [hereditary] holding [of office] — was this not Heaven’s protection, that “opening must have a precedent”, which we know-not-why-yet [happened] so?
Pre-eminent among Our [Qīng] dynasty: the Sage Ancestor Benevolent Imperial Ancestor [Kāngxī emperor], heaven-begotten and intelligent, qiánkūn combined-by-tally, his imperially-composed Shùlǐ jīngyùn 數理精藴, Lìxiàng kǎochéng 歷象考成 and other compilations, further developed the subtle-and-vague, exhausting the rectifications and variations: as the Yuèlí second-and-third equation-numbers, divided as two tables; the eclipse-changes from huángpíng xiàngxiàn (yellow-mean-image-quadrant) to báipíng xiàngxiàn (white-mean-image-quadrant); the use of the altitude-arc to determine up-down-left-right azimuth; the additions of the jiègēnfāng (borrowed-root-square, i.e. algebra) method, the explanation of the logarithmic method, [and] the explanations at the ends of the point-line-surface-and-solid sections — all matters this book had not yet been able to reach. The eight-line tables [trigonometric tables] previously took the radius as 100,000, listing each line’s value by minute; now [we] have changed the radius to 10,000,000, listing each line’s value by 10-second [intervals], using [these] for step-computation, especially short-and-quick.
As for the imperially-determined Lìxiàng kǎochéng hòubiān 歷象考成後編 — sun-and-moon use their own [orbital] heaven as ellipses; eclipses use the sun-and-moon’s two longitudes’ oblique-distance as the white-path; the apparent-motion is taken as the apparent-distance — the precision of step-computation will serve as a model for ten-thousand years. This again is what Guāngqǐ and the others could not aspire to.
However, the source from which the Shòushí-revising rectification proceeded — its source-stream truly originates in this compilation. Therefore [we] have completely recorded-and-preserved it, so that those discussing the original-form of the Western methods will have something to consult here.
Respectfully collated, Qiánlóng 46, eleventh month [December 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 window: 1629 (the Chóngzhēn 2 imperial chartering of the Lìjú under Xú Guāngqǐ and Lǐ Zhīzǎo) – 1645 (the Schall von Bell appendices added under Shùnzhì, completing the work in its received form). The principal compilation activity took place 1629–1635. Internal phases: Phase I (1629–1630): Schreck and Longobardo provide the European mathematical-astronomical foundation, Lǐ Zhīzǎo coordinates the Chinese-language drafting; both die in 1630. Phase II (1630–1633): Rho arrives in Beijing to replace Schreck; Schall arrives to assist; Xú Guāngqǐ presents successive juàn to the throne; the Wèi Wénkuí controversy is resolved with Xú’s Xuélì xiǎobiàn; Xú dies November 1633. Phase III (1633–1635): Lǐ Tiānjīng succeeds as director; final volumes presented; project formally completed in Chóngzhēn 8 (1635). Phase IV (1644–1645): Schall presents the materials to the new Shùnzhì regime; the calendar is promulgated as the Shíxiàn lì; Schall adds the two appendices Lìfǎ Xīchuán and Xīnfǎ biǎoyì.
The Chóngzhēn lìshū / Xīnfǎ suànshū is the foundational document of the post-1644 institutional integration of European mathematical astronomy into the Chinese imperial state. Its substantive contributions are immense: it is the first Chinese work to (a) systematically expound spherical trigonometry (hú sānjiǎo 弧三角); (b) introduce logarithms — although the 提要 notes that the log-fǎ jiě (explanation of logarithms) is a Kāngxī-period addition not present in the original; (c) provide European-style ephemerides for sun, moon, and planets; (d) provide the first Sino-European star catalog ordered by ecliptic coordinates; (e) introduce European astronomical instruments (including the great Beijing astronomical instruments later constructed by Verbiest and others); and (f) supply the methodological apparatus (eclipse-computation by parallax-and-refraction-correction, planetary-orbit by Tychonic geometry) that distinguished post-Chóngzhēn Chinese state astronomy from its YuánMíng predecessors.
The 提要’s framing is unusually historical-self-aware. It traces an explicit lineage: late-Míng calendrical breakdown → Wànlì-period Jesuit foundational works (Ricci, de Ursis, Dias) → 1611 Ministry-of-Rites memorial blocked → 1629 imperial chartering of the Lìjú under Xú Guāngqǐ → Chóngzhēn-period compilation in collaboration with Longobardo, Schreck, Rho, Schall → 1645 Qīng promulgation as Shíxiàn lì → Kāngxī-period synthetic refinement in Shùlǐ jīngyùn and Lìxiàng kǎochéng → Yōngzhèng-Qián-lóng-period further refinement in the Hòubiān with elliptical orbits and corrected white-path eclipse method. The Sìkù editors’ explicit acknowledgment that the post-Chóngzhēn Qīng achievements all rest on the Xīnfǎ suànshū foundation — qí yuánliú shí běn yú shì biān 其源流實本於是編 (“its source-stream truly originates in this compilation”) — is a strong testimony, by the highest editorial authority of late-imperial China, to the foundational character of the Jesuit-Chinese collaboration.
The 提要’s explicit recognition of the project’s frustration under the Míng — the parallel-bureau testing arrangement, the year-after-year confirmation of the new method’s superiority, and the failure of institutional adoption due to “court-minister factionalism” (tíngchén zhī ménhù 廷臣之門户) — is also notable: late-Qīng editorial reflection on the political dynamics that delayed the calendar’s adoption, framed as the regrettable preliminary to the Qīng dynasty’s enlightened reception of the work in 1644–1645.
Several technical observations from the 提要 deserve special note:
(1) The Kāngxī-period revision of the trigonometric tables: from radius = 100,000 (the original Lìshū convention, following Tycho’s Astronomiae) to radius = 10,000,000 (the Kāngxī-period convention, after the European trigonometric standard had shifted), with values listed at 10-arcsecond intervals. This is the first Chinese-language reference to the seven-place trigonometric tables that became the standard of Qīng mathematical astronomy.
(2) The transition to elliptical orbits: the 提要 explicitly notes that the Yōngzhèng-Qián-lóng-period Lìxiàng kǎochéng hòubiān 歷象考成後編 (1742) was the first official Chinese astronomical work to adopt elliptical solar-and-lunar orbits — a Keplerian reform that the original 1635 Chóngzhēn project had not implemented (it had used Tychonic uniform circular orbits with epicycles for the planets and lunar motion). This historical self-awareness about the gradual European-to-Chinese transmission of post-Tycho post-Kepler astronomical theory is a striking moment in the Sìkù tíyào.
(3) The introduction of algebra (jiègēnfāng 借根方, “borrowed-root-square”): the 提要 attributes this to the Kāngxī-period imperial-mathematical academy, distinguishing it from the Xīnfǎ suànshū’s purely geometric methodology. The Kāngxī-period algebra material derives from a different European source-tradition (from Antoine Thomas, S.J., and from the Algebra of Christoph Clavius, transmitted later than the original Lìshū materials).
For the Wèi Wénkuí 魏文魁 anti-reform polemics that the Xuélì xiǎobiàn refuted, see KR3f0008 (where Wèi Wénkuí is celebrated by Xíng Yúnlù as a great mathematician — the same man) and 邢雲路. For the broader Jesuit-Chinese-collaborative context, see 利瑪竇, 熊三拔, 陽瑪諾, 徐光啟, 李之藻, 李天經, 龍華民, 鄧玉函, 羅雅谷, 湯若望.
Translations and research
- Hashimoto Keizō 橋本敬造. Hō Yū-ran: Christian Mission and Calendrical Reform in Late Ming China, Kyoto: Kansai University Press, 1988. The standard study of the Chóngzhēn lì-shū project in any Western language.
- Sivin, Nathan. Granting the Seasons: The Chinese Astronomical Reform of 1280, New York: Springer, 2009 (the Shòu-shí lì whose successor the Xīnfǎ suànshū was; essential context).
- Cullen, Christopher. Heavenly Numbers: Astronomy and Authority in Early Imperial China, Oxford: Oxford University Press, 2017.
- Jami, Catherine. The Emperor’s New Mathematics: Western Learning and Imperial Authority during the Kangxi Reign (1662–1722), Oxford: Oxford University Press, 2012. The standard study of the Kāngxī-period developments built on the Xīnfǎ suànshū foundation.
- Standaert, Nicolas (ed.). Handbook of Christianity in China, vol. 1: 635–1800, Leiden: Brill, 2001 (essential reference on the Jesuit context).
- D’Elia, Pasquale M., S.J. Galileo in Cina, Roma: Pontificia Universitas Gregoriana, 1947 (treats the Galileo-Schreck correspondence and the Western sources of the Lì-shū).
- Liáng Jiāmiǎn 梁家勉, Xú Guāngqǐ nián-pǔ 徐光啟年譜, Shànghǎi: Shànghǎi Gǔjí Chūbǎnshè, 1981.
- Wáng Xiǎohóu 王肖侯, Chóngzhēn lì-shū yán-jiū 崇禎曆書研究, Shànghǎi: Shànghǎi Gǔjí Chūbǎnshè, 2017.
- Engelfriet, Peter M. Euclid in China, Sinica Leidensia 40, Leiden: Brill, 1998 (background on the Ricci-Xú collaboration that prefigured the Lì-shū project).
Other points of interest
The Xīnfǎ suànshū / Chóngzhēn lìshū is one of the few Sìkù entries whose Chinese-language transmission is doubly indirect: the underlying European astronomical theory was transmitted from Tycho (d. 1601) and Kepler (d. 1630) through their already-printed European publications (Tycho’s Progymnasmata, Kepler’s Tabulae Rudolphinae) which the Beijing Jesuits had to physically transport from Europe (Trigault’s 1618 mission with Schreck-Terrentius is the principal source); and the resulting Chinese-language synthesis was then itself revised by the Qīng imperial mathematical academy in the Kāngxī-period Shùlǐ jīngyùn and Lìxiàng kǎochéng productions. The 提要’s framing of this layered transmission as a coherent intellectual history — from late-Míng failure to Qīng triumph, with the Jesuit collaboration as the indispensable middle term — is the standard late-imperial Chinese narrative of the period 1629–1742 in mathematical astronomy.
The work’s institutional history also makes it a unique case: the Chóngzhēn lìshū was completed (1635) but not promulgated (the late Míng court continued to use the Dàtǒng lì against the increasingly strong evidence of the new method’s superiority); promulgated (1645) by a regime (the Qīng) different from the one that commissioned it; and renamed (Xīnfǎ suànshū) to disguise its Míng-dynastic origin in the post-1644 environment. The decision of the Sìkù editors, in 1781, to retain the Schall von Bell appendices intact (despite Schall’s complete identification with the Catholic mission and his late prosecution under Yáng Guāngxiān in 1664) demonstrates the Qiánlóng court’s commitment to preserving the documentary record of the Jesuit-Chinese collaboration even at the cost of theological and political tension.
Wèi Wénkuí 魏文魁 — the Mǎnchéng commoner whose conservative Lìyuán and Lìcè Xú Guāngqǐ refuted in the Xuélì xiǎobiàn (now embedded in the Xīnfǎ suànshū front matter) — is the same man whom Xíng Yúnlù 邢雲路 had celebrated in the preface to KR3f0008 Gǔjīn lǜlì kǎo (1596–1611) as “comparable to Zǔ Chōngzhī and Chén Déyī of the ancients”. The continuity of the same individual across a generation, from late-Wànlì calendar reformer (alongside Xíng) to early-Chóngzhēn anti-Western reactionary (against Xú Guāngqǐ), is one of the more telling biographical-political continuities of the late-Míng intellectual scene.