"The law of excluded middle is the logical principle in
accordance with which every proposition is either true or
false. This principle is used, in particular, whenever a proof
is made by the method of reductio ad absurdum . And it is
this principle, also, which enables us to say that the denial of
the denial of a proposition is equivalent to the assertion of
the proposition."
— Alonzo Church, "On the Law of Excluded Middle,"
Bulletin of the American Mathematical Society ,
Vol. 34, No. 1 (Jan.–Feb. 1928), pp. 75–78
It seems reasonable to define a Euclidean geometry as one describing what mathematicians now call a Euclidean space.
What sort of geometry
is the following?
Four points and six lines,
with parallel lines indicated
by being colored alike.
Consider the proposition "The finite geometry with four points and six lines is non-Euclidean."
Consider its negation. Absurd? Of course.
"Non-Euclidean," therefore, does not apply only to geometries that violate Euclid's parallel postulate.
The problem here is not with geometry, but with writings about geometry.
"In the plainest terms, non-Euclidean geometry
took something that was rather simple and straightforward
(Euclidean geometry) and made it endlessly more difficult."
Had the Greeks investigated finite geometry before Euclid came along, the reverse would be true.