On Zhu Shijie's Elimination Theory (Study of the History of Mathematics 2021)

Abstract

Zhu Shijie's siyuan shu (the four elements method) and his elimination of unknowns made a significant contribution to the history of mathematics in the world in the 14th century. The final equations given by Zhu Shijie in his book Si Yuan Yu Jian are sometimes simpler than the resultants obtained from the simultaneous equations. This shows that Zhu Shijie took additional special considerations in eliminating unknowns. In §4.1 of the present note we will discuss the second problem in Section 6 of the last chapter of Book III of Si Yuan Yu Jian , and give two possible arguments which Zhu Shijie would have used to arrive at his final equation. We will also show that it is possible to simplify his final equation for the fifth problem in the same section by using additional considerations as in the second problem. However, he did not use such arguments. This shows that Zhu Shijie did not use the additional considerations systematically. In §4.3 we will give the list of of the given conditions, the simultaneous equations, the resultant I calculated and Zhu Shijie's final equations of all problems of the section 8 of the last chapter of Book III of Si Yuan Yu Jian , which Zhu Shijie used four unknowns to solve the problems. The list shows that only the last problem of Section 8 requires to use of four unknowns, while three unknowns are sufficient for other problems.