In Universal Arithmetick (1728), Newton discussed the problem of eliminating the variable from two higher-degree equations, which was the beginning of modern elimination theory and had a great influence on later mathematicians. In this discussion, Newton listed four complicated rules of the elimination for two higher-degree equations, but did not show his deduction reasoning for these rules. The purpose of this paper is to recover Newton's deduction process of the four rules. By comparing Newton's statement on his elimination method with Euler's explanation on it, we clarify Newton's elimination operation procedure and give an explanation that is different from Euler's. Since the concluding equations are not direct results of the elimination process, we conduct the recovery in two steps. In the first step, we recover the elimination process of these rules and thus obtain the characteristic equation of each rule; in the second step, we dig into the information behind the special presentation of the concluding equations, summarize the principles by which Newton rewrote the characteristic equations, and recover the concrete operating process of how Newton obtained the four concluding equations. By means of this study, we help to unveil and understand Newton's method and technique of elimination more completely, while also evaluating his role in the development of elimination theory more accurately.
Zhao et al. (Sat,) studied this question.