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

103, 104] IS EUCLIDEAN GEOMETRY TRUE. 175
And another French geometer writes:
" We are then entitled to say that the geometry which most
closely resembles reality is the Euclidean Geometry, or at least
one which differs very little from it;... the error is too
small to be apparent in the domain of our observations and
with the aid of the instruments at our disposal.
" In a word, not only have we theoretically to adopt the
Euclidean Geometry, but in addition this geometry is physically
true." *
The matter can be put in another way. The question
whether the Euclidean Geometry is the true geometry has no
place in Geometry-the Pure Science. It has a place in
Geometry-the Applied Science. The answer to the question
-if an answer can be given-lies with the experimenter. But
his reply is inconclusive. All that he can tell us is that the
sum of the angles of any triangle that he has observedhowever great the triangle may have been-is equal to two
right angles, subject to the possible errors of observation.
To say that it is exactly two right angles is beyond his power.
One interesting point must be mentioned in conclusion.
In the Theory of Relativity, it is the Non-Euclidean Geometry
of Bolyai and Lobatschewsky which, in some ways at least,
is the more convenient. Gauss's jesting remark that he would
be rather glad if the Euclidean Geometry were not the true
geometry, because then we would have an absolute measure
of length, finds an echo in the writings of those who in these
last years have developed this new theory.t
* Hadamard, Leeons de Gdometrie elementaire, vol. i. p. 286 (Paris, 1898).
f Cf. the letter to Taurinus, quoted on p. 24. Also the letter to
Gerling given in Gauss, Werke, vol. viii. p. 169.
A similar remark is to be found in Lambert's Theorie der ParallelLinien, ~ 80; see Engel u. Stackel, loc. cit. p. 200.