华蘅芳《积较术》数学方法分析

在《积较术》中,基于独特的差分定义,华蘅芳构造了一个与Ｎｅｗｔｏｎ有限差分公式完全不同的差分体系,针对各种数表的使用,华蘅芳设计了一种”乘表相加“的计算方法。算理分析表明,这一算法与近代矩阵乘法一致。对《积较术》 中的数学思想与数学方法的分析,揭示了清末传统数学研究所的生长点及其在向近代数学转变过程的积极意义。

A STUDY ON THE MATHEMATICAL METHODS AND IDEAS INHUA HENGFANG'S JI JIAO SHU

Ji Jiao Shu (The Methods of Limited Difference) is an important mathematicswork written by Hua Hengfang (1833 - 1902). In it, Hua constructed a special system of limited difference, in which the notation of "0-side difference" (_iy_0) was a distinct characteristic.By these, series interpolation formulas were obtained, which proved to be very different fromNewton's. In order to deal with many kinds of tables, he designed an algorithm, which we nowcall "multiplying-table-and-adding". in fact, this algorithm proved the same as matrix multiplication. Furthermore, his operation of reckoning the difference inversion table (H~n_k) from thetable of x~n-difference (h~n_k) was the same as the techniques of finding inverse matrix. Based onthe above analysis, we bring to light Hua Hengfang's mathematical methods and ideas. It wasobvious that Hua's methods and ideas were the outcome of construction and mechanization,which are the very characteristics of Chinese classical mathematics. Just these played an important role in the transformation of Chinese mathematics from the classical to the modern.