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日法朔余強弱考

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Text ID
KR3fc056
Date
1795–1817
Dynasty
清

Rìfǎ shuòyú qiángruò kǎo 日法朔余強弱考

Investigation of the Stronger / Weaker Residuals in the Day-Coefficient and Lunation-Remainder by 李銳 (撰)

About the work

李銳 Lǐ Ruì’s (1768–1817) evidential mathematical study in 1 juàn of the rìfǎ 日法 (day-coefficient: the integer denominator into which a Chinese calendar partitions the mean day for purposes of fractional arithmetic) and the shuòyú 朔余 (lunation-remainder: the fractional excess of the mean synodic month over an integer number of days, expressed in rìfǎ units). The “stronger / weaker” (qiángruò 強弱) of the title refers to the sign of the residual when measured against successive trial values.

Abstract

The rìfǎ and shuòyú are the two most fundamental constants of any Chinese calendrical system. Together they determine the length of the mean synodic month, which together with the tropical-year constant determines the calendar’s intercalation pattern. Different Chinese calendrical systems adopt different rìfǎ values (the Sìfēn 四分 uses rìfǎ = 4, the Sāntǒng 三統 uses rìfǎ = 81, the Qiánxiàng 乾象 uses rìfǎ = 1457, the Sòng systems use values in the tens of thousands) — the choice reflects an arithmetic compromise between accuracy of the resulting synodic-month value and tractability of the resulting fraction.

Lǐ Ruì’s Rìfǎ shuòyú qiángruò kǎo analyses the systematic deviation (qiáng 強 “stronger” — i.e. excess; ruò 弱 “weaker” — i.e. deficient) of the resulting synodic-month values from the observed synodic-month value across the historical sequence of Chinese calendars. The study is in effect a continued-fraction-style analysis of the rational approximations to the synodic-month constant adopted by successive Chinese calendrical systems — a piece of pure number-theoretic analysis of the calendrical constants, separate from the procedural calendrical reconstructions of KR3fc051–KR3fc055.

The work shows Lǐ Ruì’s interest extending beyond reconstruction of specific calendars to the underlying theoretical structure of the Chinese calendrical tradition as a whole. Combined with the calendrical reconstructions and the equation-theoretic work of KR3fc060 Kāifāng shuō, it positions Lǐ Ruì as the most theoretically-orientated of the QiánJiā mathematical evidential scholars.

Dating: composed during Lǐ Ruì’s mature productive period. notBefore 1795; notAfter 1817.

Translations and research

  • Cullen, Christopher. 2017. The Foundations of Celestial Reckoning. London: Routledge. — Discusses the rì-fǎ concept across the early Chinese calendrical systems.
  • Sivin, Nathan. 2009. Granting the Seasons. New York: Springer.
  • Qū Ānjīng 曲安京. 2005. Zhōng-guó lì-fǎ yǔ shù-xué 中國曆法與數學. Běi-jīng: Kē-xué chū-bǎn-shè.

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