Rifa shuoyu qiangruo kao 日法朔餘强弱考

Rifa shuoyu qiangruo kao 日法朔餘强弱考, is a book on astronomy written during the high Qing period 清 (1644-1911) by Li Rui 李鋭 (d. 1817), courtesy name Shangzhi 尚之, style Sixiang 四香, from Yuanhe 元和, Jiangsu. It was finished in 1799 and is included in the literary remains of Li Rui, Lishi yishu 李氏遺書. It was reprinted several times in the 19th century. Li also wrote the books Han sifen shu zhu 漢四分術注, Han santong shu zhu 漢三統術注, Han Qianxiang shu zhu 漢乾象術注,Buxiu Song Fengyuan shu bing zhu 補修宋奉元術并注, Buxiu Song zhantian shu bing zhu 補修宋占天術并注, Zhouyi Yushi lüeli 周易虞氏略例, Zhaogao riming kao 召誥日名考, Tianyuan gougu xicao 天元句股細草, Fangczeng xinshu xicao 方程新術細草, Ceyuan haijing xicao 測圓海鏡細草 and Hushi xuanshu xicao tujie 弧矢算術細草圖解 (revised by Feng Guifen 馮桂芬, 1809-1874).

The book investigates the ancient Chinese method of calendar adjustment, known as the "Method of adjusting days" (tiaorifa 調日法). It is well known that the length of a synodic month contains a fractional remainder. The denominator of this fraction is referred to as rifa 日法. To avoid confusion with the denominator used for the length of the tropical year, the term shuorifa 朔日法 is used specifically for the synodic month. Since Liu Hong's 劉洪 (c. 130-196 CE) Qianxiang Calendar 乾象曆 of the Later Han period 後漢 (25-220 CE), the denominator for the tropical year has been separately referred to as jifa 紀法 or qifa 氣法, among other names. The numerator of the fraction is called shuoyu 朔餘. Different calendar systems have adopted different values for these fractions, but the method by which they were originally derived had long been lost, especially after the Tang dynasty's 唐 (618-907) Linde Calendar 麟德曆 (665) introduced a common denominator system. Later, decimal and ten-thousand-based fractions were adopted, further obscuring the original calculation method.

Li Rui's book investigates this issue and concludes that these values were obtained through the mathematical method of adjusting days. The process involves determining a "strong rate" and a "weak rate" (qianglü 强率, ruolü 弱率), then using specific techniques to adjust the denominator (rifa) and numerator (shuoyu) to derive the appropriate fractions. Using this principle, Li Rui examined the strength and weakness values of denominator and numerator as recorded in the books Kaiyuan zhanjing 開元占經 and Shoushi liyi 授時曆議, which document fifty-one different calendar systems. His findings showed that thirty-five of them adhered to the method of adjusting days, while sixteen did not. The discrepancies in the latter cases were attributed to three types of errors made by the calendar creators.