唐宋历法演纪上元实例及算法分析

ANALYSIS OF EXAMPLES OF "YAN JI SHANG YUAN" AND THEIR ALGORITHM IN CALENDARS OF THE TANG AND SONG DYNASTIES

唐宋历法的上元积年以演纪之法推求,称为"演纪上元"。按宋代秦九韶《数书九章》中的"演纪术",历家只需依代入法求解三个联立的同余式即将上元推定;而通常历法中从上元起算的周期多达11项,倘全部作为先决条件参与其上元的计算,则必须处理10个联立的同余式。本文首先根据史料对演纪术所列同余式组的余数选择及所得结果进行了必要的限定,然后依演纪术对明天历及"授时历议"中附列的三种积年日法进行了实例演算。结果表明,绝大多数情形下,三个同余式即将上元唯一确定,其它历法周期因此不复具备参与推算其上元的必要条件。

'Yan Ji' is a method which was employed to calculate the superior epoch (Shang Yuan) during the Tang and Song Dynasties. This method is narrated in detail in Qin Jiushao's Nine Chapters on the Mathematical Art. According to Qin's 'Yan Ji formula', there are only three congruence expressions that we have to solve one by one, but many scholars hold that all of the items in calendars which concern with the superior epoch should participate in the calculation. So under normal conditions we must settle ten congruence expressions. In this paper, we give two limits of remainders and the result first, and then elaborate our reason. On the basis of these prerequisite conditions, we demonstrate some examples of the calculation of the superior epoch in the Mingtain Calendar and On the Shoushi Calendar by Qin's method. The result shows, in most cases, that with these three congruence expressions we have no choice but to accept the superior epoch given by the calendar-makers. That means there is no chance for the other items to have a hand in the calculation of 'Yan Ji Shang Yuan'.