中国古代历法中的九服晷影算法

九服晷影算法从一行《大衍历》（公元724元）起，成为唐宋金元历法计算的组成部分。通过对这一时期九服晷影算法的系统解读，说明该算法的造术方法及特征。一行《大衍历》中采用数表算法，边冈《崇玄历》（公元892年）及其后历法均采用公式算法。王朴在《钦天历》（公元956年）中利用一行开元年间大地子午线测量结果，构造了独特的九服晷影函数，由此可推导出中国数理天文学史上第一个正切函数表达式。

Algorithm for Calculating Solar Shadows in Ancient Chinese Calendars

In ancient China,the areas far from the national capital were called by the joint name of Jiu Fu. How to calculate the length of solar shadows casted by a 8-Chi-gnomon in these areas had become a necessary work in traditional calendars after the completion of the Dayan Calendar by the Buddhist monk Yi Xing. The algorithms for calculating solar shadows in Jiu Fu were different from the others in the calendars from the Tang Dynasty to the Yuan Dynasty. Based on circumspect interpretation of relevant texts in the calendars during this preiod, the principle and characteristic of these algorithms are verified. Yi Xing used the table of solar shadows in his Dayan Calendar. Bian Gang adopted a formula of the solar shadows in his Chongxuan Calendar. In the Qintian Calendar, by taking advantage of the result of Yi Xing's meridian survey during the Kaiyuan period of the Tang Dynasty, Wang Pu proposed a particular formula of solar shadows in Jiu Fu, from which the first expression of tangent function in the history of Mathematical Astronomy can be drawn.