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象数一原

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Text ID
KR3fc069
Date
1830–1857
Dynasty
清

Xiàngshù yīyuán 象数一原

A Common Origin for Figures and Numbers by 項名達 (撰); 戴煦 (校補)

About the work

項名達 Xiàng Míngdá’s (1789–1850) principal theoretical treatise in 9 juàn, posthumously completed and edited by his student 戴煦 Dài Xù (1805–1860). The work is the major theoretical synthesis of the indigenous Chinese cyclotomic-series tradition descending from 明安圖 Míng Āntú’s Gēyuán mìlǜ jiéfǎ 割圈密率捷法 through 董佑誠 Dǒng Yòuchéng’s KR3fc068 Tújiě.

Abstract

The title Xiàngshù yīyuán (“a common origin for figures [xiàng 象] and numbers [shù 數]”) signals the work’s theoretical-philosophical orientation: the systematic demonstration that the geometric configurations (xiàng) of the circle and its inscribed-and-circumscribed polygons stand in algebraic-numerical relations (shù) expressible as infinite series, and that those series arise from a single underlying procedure — the liánbǐlì 連比例 (continued-proportion) construction.

The 9-juàn organisation systematically extends Míng Āntú’s cyclotomic series in three directions: (1) general expansion of the xiàngshù relation to cover the four principal arc-chord-sagitta-area trigonometric identities; (2) systematic derivation of the series in indigenous Chinese geometric terms (the continued-proportion procedure of cyclotomic figures); and (3) computational tabulation of the series to high orders for practical application.

Xiàng Míngdá did not live to complete the work himself. He died in 1850 with the manuscript in advanced but unfinished state. 戴煦 Dài Xù, his student and principal mathematical correspondent, took on the editorial work — completing missing computations, supplying corrections, and seeing the work through to publication. The published edition explicitly credits the 校補 (collation-and-supplementation) to Dài Xù. The collaboration is among the principal documented teacher-student mathematical lineages of the Qīng.

The Xiàngshù yīyuán represents the high theoretical synthesis of the Míng Āntú cyclotomic-series tradition. By the time of its publication, however, the European trigonometric infinite-series literature had begun to reach China directly through 李善蘭 Lǐ Shànlán’s translation programme (e.g. the translation of De Morgan’s Elements of Algebra and Loomis’s Analytical Geometry). The Xiàng – Dài work is thus the indigenous-tradition counterpart to Lǐ Shànlán’s translated material, and the two streams converged in the mathematical writing of the Tóng-zhì-era xīxué dōngjiàn 西學東漸 (“Westward-coming-eastward of Western learning”) movement.

Dating: Xiàng Míngdá lived 1789–1850. The work was composed across his mature career and completed posthumously by Dài Xù. notBefore 1830 (early productive period); notAfter 1857 (the conventional publication date under Dài Xù’s editorship; alternative dates in 1850s appear in the scholarship).

Translations and research

  • Jami, Catherine. 1990. Les méthodes rapides pour la trigonométrie et le rapport précis du cercle. Paris: Mémoires de l’Institut des Hautes Études Chinoises.
  • Tián Miǎo 田淼. 2003. Zhōng-guó shù-xué de xī-huà lì-chéng 中國數學的西化歷程. Jǐ-nán: Shān-dōng jiāo-yù chū-bǎn-shè.
  • Hán Qí 韓琦. 1991. “The Influence of Ming Antu’s Quick Methods for Determining Segment Areas…” Historia Scientiarum 1: 219–229.
  • Wú Wénjùn 吳文俊, ed. 1985. Zhōng-guó shù-xué shǐ dà-xì 中國數學史大系, vol. 7–8.

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