Taylor's expansion and its applications to analysis

In the following pages, I have attempted to develop Taylor's expansion such that the reader may see his method, the modern treatment, and the resulting applications. The greater part of the treatise deals with Taylor's expansion for a function of one real variable and the applications which arises from that expansion. In accordance with this aim, chapter one is an outline of Taylor's method; chapter two deals with the modern method of developing Taylor's expansion for a function of one real variable; and chapter three deals with the applications of this expansion. I have further dealt briefly, in chapters four and five, with the expansion for a function of several variables, and the expansion for a function of a complex variable, respectively