Axiomatic and Coordinate Geometry

At some point between high school and college we first make the transition between Euclidean (or synthetic) geometry and co-ordinate (or analytic) geometry. Later, during graduate studies we are introduced to differential geometry of many dimensions. The justification given in the first instance is that coordinates are a natural outcome of the axioms of Euclidean geometry; and in the second case because Riemannian geometry is much more general than axiomatic non-Euclidean geometry. In this expository account we examine these two justifications.