Non-Euclidian Geometry

Non-Euclidean Geometry is now recognized as an important branch of Mathematics. Those who teach Geometry should have some knowledge of this subject,and all who are interested in Mathematics will find much to stimulate them andmuch for them to enjoy in the novel results and views that it presents.

Excerpt:

This book is an attempt to give a simple and direct account of the Non-Euclidean Geometry, and one which presupposes but little knowledge of Mathematics.

The first three chapters assume a knowledge of only Plane and SolidGeometry and Trigonometry, and the entire book can be read by one who hastaken the mathematical courses commonly given in our colleges.

No special claim to originality can be made for what is published here. Thepropositions have long been established, and in various ways. Some of the proofsmay be new, but others, as already given by writers on this subject, could not beimproved. These have come to me chiefly through the translations of ProfessorGeorge Bruce Halsted of the University of Texas.

The axioms of Geometry were formerly regarded as laws of thought which anintelligent mind could neither deny nor investigate. Not only were the axiomsto which we have been accustomed found to agree with our experience, but itwas believed that we could not reason on the supposition that any of them arenot true, it has been shown, however, that it is possible to take a set of axioms,wholly or in part contradicting those of Euclid, and build up a Geometry asconsistent as his.

We shall give the two most important Non-Euclidean Geometries. In thesethe axioms and definitions are taken as in Euclid, with the exception of thoserelating to parallel lines. Omitting the axiom on parallels, we are led to threehypotheses; one of these establishes the Geometry of Euclid, while each of theother two gives us a series of propositions both interesting and useful. Indeed, aslong as we can examine but a limited portion of the universe, it is not possibleto prove that the system of Euclid is true, rather than one of the two Non-Euclidean Geometries which we are about to describe.

We shall adopt an arrangement which enables us to prove first the propositionscommon to the three Geometries, then to produce a series of propositionsand the trigonometrical formulæ for each of the two Geometries which differfrom that of Euclid, and by analytical methods to derive some of their moststriking properties.

We do not propose to investigate directly the foundations of Geometry, noreven to point out all of the assumptions which have been made, consciously orunconsciously, in this study. Leaving undisturbed that which these Geometrieshave in common, we are free to fix our attention upon their differences. By aconcrete exposition it may be possible to learn more of the nature of Geometrythan from abstract theory alone.