Between 1822 and 1823, the Qing dynasty official Fu Jiuyuan (傅九渊, 1791—1845) completed a 4-volume work on mathematics, Youbuwei Zhai Suanxue (有不为斋算学, Mathematics Youbuwei Studio). Volume 1 explains zhaocha shu (招差术, finite difference method) while Volume 2 gives examples of finite difference calculations. These two volumes represent the first specialized work to discuss this method. Fu first clearly defines core concepts such as ji (积, term of a sequence), jicha (积差, the difference between the first and last terms of a sequence), and zhaocha (招差, expressing a certain term using the differences of a sequence). He points out that zhaocha shu is an algorithm that linearly expresses the general term (a polynomial) of a higher-order arithmetic sequence using the first term of the sequence and the first terms of its various order differences. Subsequently, he clearly summarizes the corresponding relationship between the coefficients of the first term in this expression and the numbers in the corresponding horizontal row of Jia Xian's Triangle (贾宪三角形, Jia Xian's Diagram similar to Pascal's Triangle), and proposes the mutual calculation algorithm for these numbers as well as the basic stacking properties of Jia Xian's Triangle. Finally, he explicates the general term algorithm and summation algorithm for higher-order arithmetic sequences. This achievement is equivalent to recognizing that zhaocha shu is essentially an algorithm that converts polynomials into difference expressions. Thus, Fu Jiuyuan accurately, comprehensively and clearly elaborates the mathematical principles of zhaocha shu.
GAO et al. (Mon,) studied this question.