The monograph is divided into nine chapters. The first chapter gives the preliminaries: sets, an informal (as opposed to formal) introduction to logic and some basic facts about real numbers. The nonstandard extension of the field of real numbers shows its face in Chapter 2, where the author calls such an extension the ”Euclidean line (this should not be confused with the concept of Euclidean fields, those ordered fields whose positive elements have a square root). He axiomatizes as a field containing and also an infinitely big elements and explores the first ideas of nonstandard calculus and gives the first examples of the Transfer Principle. The construction of such a field is done via ultrapowers in the last chapter. The remaining chapters, Chapter 3 through Chapter 8, treat the concepts of calculus in this setting: continuity, derivatives, integration, some theorems of differential calculus, some special functions, and series. The book is written in a reader-friendly style and somewhat reflects the humor of the author. It can be read as an introduction to nonstandard analysis by an undergraduate student who has been exposed to a first course of calculus. [...]