This thesis is based on M. J. Greenberg's article "Old and New Results in the Foundation of Elementary Plane Euclidean and Non-Euclidean Geometries" (American Mathematical Monthly , Vol 117, No 3 pp. 198-219). The aim of this thesis is to give a more complete description of some of the interesting topics in this article. We will start with Hilbert's axioms and Euclid's propositions, and then focus on hyperbolic geometry. We will proceed to give a complete proof of the uniformity theorem by using Saccheri's quadrilateral. Further, implication relations between the axioms and statements which can eliminate the obtuse angle hypothesis. Lastly, we shall discuss the famous mathematic problem of "squaring the circle" in a hyperbolic plane to Fermat primes, based on W. Jagy's discovery.