Mathematical philosophy, a study of fate and freedom; lectures for educated laymen, by Cassius J. Keyser.

NON-EUCLIDEAN GEOMETRIES 355
dress Riemann indicated the possibility of constructing
a geometry upon a basis of postulates containing the assumption that
No two lines of a plane are parallel; or, what is
equivalent, every two lines of a plane intersect.
This no-parallel assumption contradicts not only the
one-parallel assumption of Euclid but another assumption
of his,-a tacit assumption,-namely, that a line has infinite extent (see propositions I6 and 28, Book I, in
Heath's edition of the Elements). The remaining assumptions of Euclid are retained by Riemann.
We have now before us three postulate systems,one of them Greek,-one of them Russian,-one of them
German. Upon them have been erected and now stand
three geometries, one of them Euclidean, the other two
non-Euclidean; of these two, the former is often called
Lobachevskian, and the latter Riemannian. For a good
reason, which I will not pause here to explain, Professor
Felix Klein has called the three geometries respectively
Parabolic, Hyperbolic, and Elliptic, descriptions that are
now in common use.
With the elements of the parabolic geometry you are
familiar; with those of the other two varieties you are
presumably not acquainted. The hyperbolic and elliptic
geometries have been built up by various methods, elementary and more advanced, pure and analytic. I credit
you with having curiosity to see how the building can be
done by the familiar elementary processes of ordinary
geometry. To apply them here would detain us too long;
but if you have the curiosity, you can gratify it by reading
the previously mentioned essay of Professor F. S. Woods
on "Non-Ecçlidean Geornetry" (found in Monographs