The aim of the present exposition is to discuss the Pythagorean theorem and the basic facts of vector geometry in a variety of mathematical and physical contexts, and to point out the significance of these notions in the special theory of relativity.

The Pythagorean theorem has suffered the same fate that so many basic mathematical facts have suffered in the course of the history of mathematics. At first, these facts were surprising when they were discovered and deep in that they required original inventive proofs. In the course of time such facts were placed into a conceptual framework in which they could be derived by more or less routine deductions; finally, in a new axiomatic arrangement of this framework, these facts were reduced to serve simply as definitions. Still, this need not have meant reduction to insignificance. What had become merely a definition may have been brought alive and made effective as a guiding principle in the development of new branches of mathematics. It is one of our aims to show that just this process describes the life cycle of the Pythagorean theorem.

In the first chapter of this exposition we begin by discussing one of the simplest proofs of the Pythagorean theorem within the framework of Euclidean geometry, and then we present a less frequently used proof.