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

NON-EUCLIDEAN GEOMETRY.
CHAPTER I.
THE PARALLEL POSTULATE, AND THE WORK OF
SACCHERI, LEGENDRE AND GAUSS.
~ 1. By the term Non-Euclidean Geometry we understand
a system of Geometry built up without the aid of the Euclidean
Parallel Hypothesis, while it contains an assumption as to
parallels incompatible with that of Euclid.
The discovery that such Non-Euclidean Geometries are
logically possible was a result of the attempts to deduce
Euclid's Parallel Hypothesis from the other assumptions
which form the foundation of his Elements of Geometry.
It will be remembered that he defines Parallel Lines as follows:
Parallel straight lines are straight lines which, being in the
same plane and being produced indefinitely in both directions,
do not meet one another in either direction.*
Then in I. 27 he proves that
If a straight line falling on two straight lines make the alternate angles equal to one another, the straight lines will be parallel
to one another.
And in I. 28 that
If a straight line falling on two straight lines make the exterior
angle equal to the interior and opposite angle on the same side,
*Here and in other places where the text of Euclid's Elements is
quoted, the rendering in Heath's Edition (Cambridge, 1908) is adopted.
This most important treatise will be cited below as Heath's Euclid.
N.-E.G. A