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

18, 19, 20]

RIEMANN

39

assume independence of bodies from position, and therefore
ascribe to space constant curvature, it must necessarily be
finite, provided this curvature has ever so small a positive
value." *
~20. Riemann, therefore, substituted for the hypothesis
that the straight line is infinite, the more general one that it is
unbounded. With this assumption the Hypothesis of the
Obtuse Angle need not be rejected. Indeed the argument which
led Saccheri, Legendre, and the others to reject that hypothesis
depended upon the theorem of the external angle (I. 16).- In
the proof of this theorem it is assumed that the straight line is
infinite.
The Hypothesis of the Obtuse Angle being available, another
Non-Euclidean Geometry appeared. The importance of this
new Geometry was first brought to light, when the ideas of
the Non-Euclidean Geometry were considered in their bearing
upon Projective Geometry.
A convenient nomenclature was introduced by Klein.t He
called the three geometries Hyperbolic, Elliptic, or Parabolic,
according as the two infinitely distant points on a straight
line are real, imaginary, or coincident. The first case we meet
'in the Geometry of Lobatschewsky and Bolyai; the second
in the Geometry of Riemann; the third in the Geometry of
Euclid. These names are now generally adopted, and the
different Non-Euclidean Geometries will be referred to below
by these terms.
It is evident that at this stage the development of the NonEuclidean Geometries passes beyond the confines of Elementary
Geometry. For that reason the Elliptic Geometry will not
receive the same treatment in this book as the simpler Hyperbolic Geometry. Also it should perhaps be pointed out herethe question will meet us again later-that the Elliptic Geometry really contains two separate cases, and that probably only
one of these was in the mind of Riemann. The twofold nature
of this Geometry was discovered by Klein.
*This quotation is taken from Clifford's translation of Riemann's
nmemoir (Nature, vol. viii. 1873). The surface of a sphere is unbounded:
it is not infinite. A two-dimensional being moving on the surface of a
sphere could walk always on and on without being brought to a stop.
tCf. Klein, "Uber die sogenannte Nicht-Euklidische Geometrie,"
Math. Ann. vol. iv. p. 577 (1871), and a paper in Math. Ann. vol. vi.
Also Bonola, loc. cit. English translation, App. iv.