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

38

NON-EUCLIDEAN GEOMETRY

[CH. II.

references to the works of Lobatschewsky and Bolyai showed
the mathematicians of that day in what esteem Gauss had held
these two still unknown and obscure names. Soon afterwards,
thanks chiefly to Lobatschewsky's works, and to the labours of
some well-known French, German, and Italian geometers, the
Non-Euclidean Geometry, which Bolyai and Lobatschewsky
had discovered and developed, began to receive full recognition.
To every student of the Foundations of Geometry their names
and their work are now equally familiar.
~19. The Work of Riemann (1826-1866).
The later development of Non-Euclidean Geometry is due
chiefly to Riemann, another Professor of Mathematics at
Gottingen. His views are to be found in his celebrated memoir:
Uber die Hypothesen welche der Geomnetrie zu Grunde liegen. This
paper was read by Riemann to the Philosophical Faculty at
Gottingen in 1854 as his Habilitationsschrift, before an audience
not composed solely of mathematicians. For this reason it
does not contain much analysis, and the conceptions introduced
are mostly of an intuitive character. The paper itself was not
published till 1866, after the death of the author; and the
developments of the Non-Euclidean Geometry due to it are
mostly the work of later hands.
Riemann regarded the postulate that the straight line is
infinite-adopted by all the other mathematicians who had
devoted themselves to the study of the Foundations of Geometry-as a postulate which was as fit a subject for discussion as
the Parallel Postulate. What he held as beyond dispute was
the unboundedness of space. The difference between the infinite and unbounded he puts in the following words:
" In the extension of space construction to the infinitely
great, we must distinguish between unboundedness and infinite
extent; the former belongs to the extent relations; the latter
to the measure relations. That space is an unbounded threefold manifoldness is an assumption which is developed by
every conception of the outer world; according to which every
instant the region of real perception is completed, and the
possible positions of a sought object are constructed, and which
by these applications is for ever confirming itself. The unboundedness of space possesses in this way a greater empirical
certainty than any external experience, but its infinite extent
by no means follows from this; on the other hand, if we