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

358 MATHEMATICAL PHILOSOPHY
that, if Euclidean geometry is self-consistent, then each of
the non-Euclidean geometries is self-consistent. And this
has been done by various mathematicians in various ways.
It has been done very simply by Henri Poincaré in his
widely known Science and Hypothesis; and it has been
done still better, in the sense of greater detail, in the previously cited Elementare Geometrie of Weber and Wellstein. Suffice it here to say,-for I am going to leave it
to you to examine the proofs,-that the principle or the
trick involved in them is that of showing the postulate
systems of Phe non-Euclidean geometries to be each of
them satisfied by suitably selected classes of geometric
entities found in Euclidean geometry. So, you see, if
the non-Euclidean geometries have any unsoundness in
them, there is a corresponding unsoundness in Euclidean
geometry. In respect of soundness-inner consistencyself-compatibility-logical concordance among the parts
of each-the three geometries are on exactly the same
level, and the level is the highest that man has attained.
The three doctrines are equally legitimate children of one
spirit,-the geometrizing spirit, which Plato thought
divine,-and they are immortal. Work inspired and approved by the muse of intellectual harmony can not perish-it is everlasting.
Another question is: Are these geometries true?
They are true in the sense in which truth resides in a
body of propositions of which some are mutually compatible premises and the rest are inevitable consequences
thereof, enchained thereto by the binding threads of logical fate, which is changeless and timeless. Such truth,
however, though it is ineffably precious, is only a quality
or an aspect of that inner consistency which gives an autonomous body of propositions its peculiar beauty, pure,