The elements of non-Euclidean plane geometry and trigonometry, by H. S. Carslaw.

vi

PREFACE

In Chapters VI.-VII. a similar treatment, though in less
detail, is given for the Elliptic Geometry.
Chapter VIII. deals with Poincare's representation of the
Non-Euclidean Geometries by the geometry of the families
of circles orthogonal or diametral to a fixed circle. From
these representations an elementary proof of the impossibility
of proving Euclid's Parallel Postulate can be obtained, and
they throw fresh light upon the Non-Euclidean Geometries
themselves.
This little book could never have been written had it not
been for the work of Bonola. It was from him that I first
learnt that an elementary treatment of the subject was possible.
Both to his historical work, an English translation of which
I had the privilege of undertaking, and to his article in Enriques'
Questioni riguardanti la geometria elementare, especially in
its extended form in the German edition of that work, this
book owes a very great deal.
The other writers on the same subject to whom I am most
indebted are Liebmann and Stickel. The treatment of Plane
Hyperbolic Trigonometry is due to Liebmann; and to the
second edition of his well-known Nichteuklidische Geometrie,
as well as to his original papers, most of which he has sent to
me as they appeared, I am much indebted. A similar acknowledgment is due to Stickel. When he learnt that I was engaged
on this work, I received from him, in the most generous way,
a set of all his papers on the subject, many of which were
inaccessible to me in Australia; and the gift of a copy of his
book on Wolfgang and Johann Bolyai, immediately on its
publication, allowed me to make some use of his final account
of the discovery of the Hyperbolic Geometry in reading my
proofs.
Other acknowledgments will be found in their proper place
in the text. However, I would mention here the frequent
use I have made of Halsted's work and of the Bibliography
of Sommerville; also the assistance which I have received