Introduction

This introductory chapter indicates the role of analogies that Einstein’s addition law of relativistically admissible velocities captures. The story of the book, unfolded here, begins in Chapter 2 with the introduction of a new look at Einstein addition and the way it gives rise to the novel algebraic structures known as gyrogroups and gyrovector spaces. The aim of this introductory chapter is to briefl y illustrate the use of gyrovector spaces in the service of analytic hyperbolic geometry [118], in full analogy with the common use of vector spaces in the service of analytic Euclidean geometry.