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割圈密率捷法

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Text ID
KR3fc037
Date
1730–1775
Dynasty
清

Gēyuán mìlǜ jiéfǎ 割圈密率捷法

Quick Method of Circle-Cutting Refined Ratios (Trigonometric Power Series) by 安圖 (Mínggatu, 撰) and 陳際新 (續, completed posthumously)

About the work

The principal Chinese-language eighteenth-century mathematical treatise, in 6 juàn — the systematic exposition of the trigonometric power-series expansions transmitted to 安圖 Mínggatu (d. 1765) by the Jesuit mathematician Pierre Jartoux (Dù Démeǐ 杜德美, 1668–1720) during their joint Bureau-of-Astronomy work in the 1710s. Incomplete at Mínggatu’s death in 1765; put through final editing and publication by his pupil 陳際新 Chén Jìxīn and others in the closing decade of Qiánlóng (c. 1775). The catalog records “extent: 6卷” and lists co-authorship 安圖[清]陳際新等續.

Abstract

The work is the foundational document of Chinese-language analytical mathematics — the first systematic presentation in Chinese of the infinite-series methods (the power-series for sine, cosine, arc-sine, and related inverse functions) that constituted the early stages of European calculus. Pierre Jartoux had received the relevant series from his European correspondents (these are essentially the Newton-Leibniz series for trigonometric functions, transmitted to Jartoux through his Jesuit network) and had communicated three principal series to Mínggatu: the series for arc-length in terms of chord, the series for chord in terms of arc, and the series for zhèngshǐ 正矢 (versed sine) in terms of arc. Jartoux had stated the series without proof.

Mínggatu’s substantive contribution was twofold:

(a) Combinatorial proofs: Mínggatu supplied detailed combinatorial proofs for each series, working in a style adapted from the 朱世傑 Zhū Shìjié duǒjī 垛積 figurate-summation tradition of the Yuán. His proofs proceed by interpreting the series coefficients as combinatorial-figurate sums (sums of products of integers from successive natural-number sequences), then establishing the series identity by direct combinatorial reasoning. This is the principal substantive original contribution of Chinese mathematics in the eighteenth century, and it represents a remarkable indigenous theoretical-mathematical achievement.

(b) Systematic extension: where Jartoux had supplied three series, Mínggatu systematically extended the framework to nine related trigonometric series (the jiǔ shù 九術 “nine procedures” enumerated in the work): series for the arc in terms of chord, the chord in terms of arc, the versed-sine in terms of arc, the chord in terms of versed-sine, etc. — covering essentially all the elementary trigonometric inverse relations.

The work is structured as: juàn 1, the basic combinatorial-figurate methodology (essentially restating the Zhū Shìjié duǒjī gē material in updated form, with the new series as immediate applications); juàn 2–6, systematic derivation of each of the nine series with full combinatorial proof.

The work is remarkable on several counts. Substantively, it is the most theoretically-sophisticated Chinese mathematical work of the eighteenth century — the only Chinese mathematical production of that period to make a substantive original contribution to mathematical theory rather than primarily exposition of imported European content. Methodologically, it represents the most successful fusion of indigenous and imported mathematical traditions in pre-modern China: where most Sino-European mathematical productions of the period work primarily within one or the other tradition, Mínggatu’s work genuinely combines them — taking the imported series-expansion results and grounding them in indigenous combinatorial-figurate methodology. Historically, it shows that the late-imperial Mongol-Manchu component of the Qīng mathematical establishment was not merely receptive to imported European learning but capable of substantive original work integrating European and East-Asian methodologies.

Dating: Mínggatu’s productive period at the Bureau extends from c. 1712 to his death in 1765; the bulk of the Gēyuán mìlǜ jiéfǎ was composed in the middle decades of that period, with completion by Chén Jìxīn c. 1775. NotBefore set at 1730 (allowing for Mínggatu’s mature productive period); notAfter at 1775 (Chén Jìxīn’s editorial completion).

Translations and research

  • Jami, Catherine. 1990. “Une histoire chinoise du ‘nombre π’.” Archives Internationales d’Histoire des Sciences 38: 39–50. — Treats the Gē-yuán mì-lǜ jié-fǎ in the context of the π-value calculation.
  • Jami, Catherine. 2011. The Emperor’s New Mathematics. Oxford: Oxford University Press. — Substantial treatment of Mínggatu’s work in the imperial-mathematical context.
  • Hé Bǐngyù 何丙郁. 1977. Les systèmes d’équations polynomes dans le Siyuan yujian. Paris: Collège de France. — Treats the relation between Mínggatu’s combinatorial methods and the Zhū Shìjié duǒ-jī tradition.
  • Liú Dùn 劉鈍, ed. 1998. Suàn-jīng shí-shū 算經十書 (modern edition). — Includes survey discussion.
  • Wú Wénjùn 吳文俊, ed. 1985. Zhōng-guó shù-xué shǐ dà-xì 中國數學史大系, vol. 7. — Standard Chinese reference.
  • Roegel, Denis. 2010. “A reconstruction of Mei Wending’s tables of the sun, the moon, and the planets.” History of Mathematics Working Papers 26. — Context for the broader Kāng-xī mathematical-astronomical project.

Other points of interest

The Gēyuán mìlǜ jiéfǎ is the unique substantive original mathematical contribution from the eighteenth-century Chinese mathematical establishment — and remarkably, it comes from a Mongol-banner mathematician working at the imperial Bureau of Astronomy rather than from the Hàn-Chinese mathematical scholars who dominated the literary record. The work’s existence is one of the strongest pieces of documentary evidence against the conventional view that eighteenth-century Chinese mathematics was a period of stagnation; in the Gēyuán mìlǜ jiéfǎ the most theoretically-advanced mathematics of the eighteenth century was being done in Beijing as well as in Europe.

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