Recently I have decided to capitulate and adopt Isaacs, which shuns both axiomatics and hyperbolic geometry in favor of actual problem solving and construction problems in standard Euclidean geometry.
It is important to note, however, that the extra dimension, though curled up, was still Euclidean in nature.
If all accelerated systems are equivalent, then Euclidean geometry cannot hold in all of them.
It seemed to me that I could do some useful work in giving the student a historical perspective and in showing how the multitude of abstract concepts have arisen and are present in Euclidean spaces.
He was one of the earliest mathematicians to demonstrate that the ordinary experience of Euclidean concepts can be extended meaningfully beyond geometry into the idealised constructions of more complex abstract mathematics.
Similarly, an eliminative structuralist account of real analysis and Euclidean geometry requires a background ontology whose cardinality is at least that of the continuum.
The falseness of the idea of principle, is typified by a Cartesian or Euclidean geometry.
In 1869, after Beltrami's letter… he realized he had made a mistake: the empirical concept of a rigid body and mathematics alone were not enough to characterize Euclidean geometry.
The approach that concentrates on non-Euclidean geometry is ideal for students who already have a mastery of Euclidean geometry, but it cannot replace such a mastery.
It reduced the problem of consistency of the axioms of non-Euclidean geometry to that of the consistency of the axioms of Euclidean geometry.