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

PREFACE
IN this little book I have attempted to treat the Elements
of Non-Euclidean Plane Geometry and Trigonometry in such
a way as to prove useful to teachers of Elementary Geometry
in schools and colleges. Recent changes in the teaching of
Geometry in England and America have made it more.than
ever necessary that the teachers should have some knowledge
of the hypotheses on which Euclidean Geometry is built, and
especially of that hypothesis on which Euclid's Theory of
Parallels rests. The historical treatment of the Theory
of Parallels leads naturally to a discussion of the Non-Euclidean
Geometries; and it is only when the logical possibility of these
Non-Euclidean Geometries is properly understood that a
teacher is entitled to form an independent opinion upon the
teaching of Elementary Geometry.
The first two chapters of this book are devoted to a short
discussion of the most important of the attempts to prove
Euclid's Parallel Postulate, and to a description of the work
of the founders of Non-Euclidean Geometry, Bolyai, Lobatschewsky and Riemann.
In Chapters III.-V. the Non-Euclidean Geometry of Bolyai
and Lobatschewsky, now known as the Hyperbolic Geometry,
is developed in a systematic manner. The feature of this
treatment is that in Chapter III. no use is made of the Principle
of Continuity, and that both the Geometry and the Trigonometry of the Hyperbolic Plane are built up without the use
of Solid Geometry.