中国古代历法中的三次内插法

CUBIC INTERPOLATION IN ANCIENT CHINESE CALENDARS

该文根据《天文大成管窥辑要》中的史料，发现边同在其《崇玄历》（８９２年）中创立的晷影公式──中国历法史上第一例三次函数，是通过今影差变化与自变量平方的比值为某个等差数列而构造出来的，与过去认为的三次内插法无关；王恂、郭守敬在《授时历》（１２８０年）创立的平立定三差算法，则是通过对插值函数的降阶，将问题转化为一般的二次内插公式的构造，前者可能受到了边冈立方相减相乘算法的启发，后者则与刘焯的二次插值算法一脉相承。

The Tianwen Dacheng Guankui Jiyao (1653) is an edited astronomical historical record which includes the construction method of some important algorithms used in ancient Chinese calendars. The solar shadow algorithm in Bian Gang's Chongxuan calendar(892),which regards the cubic interpolation method to have appeared in the Tang Dynasty,was the first known cubic function in China. By analysing its content,a constructionmethod for the solar shadow algorithm permits the ratio between the difference value of twosolar shadows and the square of independent variables to be an arithmetic sequence. It hasnothing to do with cubic interpolation. The algorithm of cubic interpolation created byWang Xun and Guo Shoujing in their Shoushi calendar (1280) is constructed as follows:first reduce the order of the interpolation function to be constructed, and then use a methodsimilar to the construction of quadratic function devised by Liu Zhuo in his Huangji calendar (600). The first step could have been inspired by Bian Gang's solar shadow algorithm.