Overview

This introductory chapter presents a few hyperbolic gems from the book to amaze both the uninitiated and the practicing expert. The actual study of analytic hyperbolic geometry, thus, begins in Chap. 2.

Geometry, according to Herodotus, and the Greek derivation of the word, had its origin in Egypt in the mensuration of land, and fixing of boundaries necessitated by the repeated inundations of the Nile. It consisted at first of isolated facts of observation and crude rules for calculation until it came under the influence of Greek thought. Following the introduction of geometry from Egypt to Greece by Thales of Miletus, 640 546 B.C., geometric objects were abstracted, thus paving the way for attempts to give geometry a connected and logical presentation. The most famous of these attempts is that of Euclid, about 300 B.C. [Sommerville (1914), p. 1].

According to the Euclid parallel postulate, given a line L and a point P not on L there is one and only one line L' which contains P and is parallel to L. Euclid's parallel postulate does not seem as intuitive as his other axioms. Hence, it was felt for many centuries that it ought to be possible to find a way of proving it from more intuitive axioms. The history of the study of parallels is full of reproaches against the lack of self-evidence of the Euclid parallel postulate. According to Sommerville [Sommerville (1914), p. 3], Sir Henry Savile referred to it as one...