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

174

NON-EUCLIDEAN GEOMETRY

[ca. VIII.

is conformal, that the sum of the angles of a nominal triangle
in this geometry is greater than two right angles. (Cf. ~ 78 (3).)
However, the metrical properties of this geometry cannot
be treated so easily as were the corresponding properties in
the geometry of the system of circles cutting the fundamental
circle orthogonally. The same argument to a certain extent
applies, but in the definition of nominal lengths the intersections with an imaginary circle have to be taken. It should
be added that in the extension to solid geometry the system
of spheres cutting a fixed sphere diametrally has to be
employed.
The fuller discussion of this nominal geometry will not be
undertaken here. If it is desired to establish the fact that
no contradiction could appear in the Elliptic Geometry,
however far that geometry were developed, there are simpler
methods available than this one. The case of the Hyperbolic
Geometry was discussed in detail, because it offered so
elementary a demonstration of the impossibility of proving
the Parallel Postulate of Euclid.
~ 104. We have already quoted some remarks of Bolyai's
on the question of whether the Euclidean or the Non-Euclidean
Geometry is the true geometry.* We shall conclude this
presentation of our subject with two quotations from modern
geometers on the same topic:
"What then," says Poincare, "are we to think of the
question: Is Euclidean Geometry true? It has no meaning.
We might as well ask if the metric system is true, and if the
old weights and measures are false; if Cartesian coordinates
are true and polar coordinates false. One geometry cannot
be more true than another; it can only be more convenient.
Now, Euclidean Geometry is, and will remain, the most
convenient: first, because it is the simplest, and it is so not
only because of our mental habits or because of the kind of
intuition that we have of Euclidean space; it is the simplest
in itself, just as a polynomial of the first degree is simpler than
a polynomial of the second degree; secondly, because it
sufficiently agrees with the properties of natural solids, those
bodies which we compare and measure by means of our senses.-t
* Cf. ~ 15.
t Poincare, La Science et M'Hypothese. English translation, p. 50.