But they constitute the substance of non-Euclidean geometry; they are its blood and sinews.
In their non-Euclidean geometry the part is always greater than the whole.
For instance, we might know that in non-Euclidean geometries, parallels meet.
It involves the theory of non-Euclidean geometry, Euclid's postulate of parallels being used in proving this theorem.
What is interesting, however, is not his fallacious conclusion, but the non-Euclidean results which he obtains in the process.
A special application of his theory of continuous groups was to the general problem of non-Euclidean geometry.
Permit me for brevity to call such a movement a non-Euclidean displacement.
Lobachevski has proved not, by creating non-Euclidean geometry.
On this postulate hang all the "law and the prophets" of the non-Euclidean Geometry.
But it is not enough that the Euclidean (or non-Euclidean) geometry can never be directly contradicted by experience.
Relating to any of several modern geometries that are based on a set of postulates other than the set proposed by Euclid, especially one in which all of the postulates of Euclidean geometry hold except the parallel postulate. Compare Euclidean.