Is Parallel Postulate Necessary?

Abstract

As a much later addition to the original Euclidean geometry, the parallel postulate distinguishes non-Euclidean geometries from Euclidean geometry. This paper will show that the parallel postulate is unnecessary because the 4th Euclidean axiom can already achieve the same goal. Furthermore, using the 4th Euclidean axiom can measure space curvature locally on manifold, while using the parallel postulate cannot.