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...Oldandnewresultsinthefoundationsof elementary plane euclidean and non geometries marvinjaygreenberg by geometry i mean the of lines circles straight edge compass constructions in both planes an axiomatic description it is sections this survey highlights some foundational history interesting recent discoveries that deserve to be better known suchasthehierarchiesofaxiomsystems aristotle saxiomasa missinglink bolyai s discovery proved generalized william jagy relationship circle squaring a hyperbolic fermat primes undecidability incom pleteness consistency much more main theme what hilbert called purity methods proof exemplied his early twentieth century successors works on foundations axiomaticdevelopment viewpoint euclid elements was rst presentation mathemat ics based ve postulates plus common notions wasn t until end nineteenth rigorous revisions axiomatics were presented lling many gaps denitions proofs revision with great est inuence david starting which will discussed below not onl...