中国古代日食食差算法的原理

Revisit to the Solar Eclipse Theory in Ancient China

日食的食差是月亮的视差对日食食甚时刻月亮到黄白交点之距离的修正值。从形式上看，其理论模型与中国古代的传统算法几乎没有任何相似之处。为了与理论算法进行比较，成功地将薮内清建立的理论模型简化为以太阳的黄经与时角为自变量的二元函数；同时，重构了传统算法的数学模型。在此基础上，比较了传统算法与理论算法的异同，得出的结论是：两者的数学模型是相似的、计算结果是近似的。以现代科学的标准来看，中国古代日食食差算法的数学模型是合理的、有效的。

In his Xuanming Li (822 AD), Xu Ang invented an algorithm for calculating the deviation caused by lunar parallax at the middle time of solar eclipse. The deviation of the distance between the Moon and the node of the ecliptic and the lunar orbit was called shicha which consisted of two independent functions, named qicha and kecha. After Xu Ang, shicha became a core algorithm of solar eclipse theory in every traditional Chinese calendar-making system. In his Ph. D thesis(1944) ,Professor Yabuuti Kiyoshi(1906-2000)established a theoretical system for calculating such a deviation of solar eclipse. Since his theoretical model is very complex, and quite different from that in Chinese calendar-making system, people still cannot make sense what the real meaning of Chinese deviation is. Following the ancient Chinese mode of thinking, according to a series of mathematical demonstration, the theoretical formula of this deviation is simplified to a function of two independent variables, longitude and hour angle of the Sun. These are variables of the function of qicha and kecha in traditional Chinese mathematical astronomy. On the other hand, based on the traditional algorithm, a mathematical model of Chinese deviation is also set up. Then, the similarities and differences between the theoretical and traditional methods are compared in detail. The result drawn from the comparison shows that the mathematical models are similar, and calculations are approximate between the two. The real meaning of Chinese deviation is now brought to light. Two conclusions, therefore, come out: 1. From the standard of modern science, the model of Chinese deviation is reasonable; 2. The algorithm based on this model is not only effective, but also fairly precise.