the classical theory of euclidean geometry is much deeper and more difficult to do properly, i.e.with full proofs, than is usually thought. extremely few people ahve ever seen a thorough development of euclidean geometry, which i find quite beautiful and stimulating.
the best soiurce is euclid himself, but it helps to have a guide, like the marvellous book of hartshorne: geometry, euclid and beyond.
the theory is actually far more complkex than that of the real numbers, as there s no need to assume the dedekind axiom. thus euclidean geometry works with subfields of the reals, closed under certain operations corresponding making various geometrical constructions.
hilberts axiomatic foundations are perhaps the best, where he explicitlky states he will not use dedekinds axiom in order to be able to make conclusions about numbers from th geometry, instead of vice versa.
the system of birkhoff does assume each line is equivalent to the reals, but birkhoff is not doing strictly euclidean geometry, since he is assuming also a unit. thus the group of automorphisms in birkhoff's system excludes scaling.
i have been very challenged and enlightened to present merely a basic course in neutral and euclidean geometry this semester, with due attention to all details of proof. in particular it is interesting to keep straight which construtions and theorems are true withiout the parallel postulate, and hence hold in hyperbolic geometry.for instance it is known the concurrence theorem for medians of triangles is true withut the parallel postulate, but the noly proof i have been shown, uses the riemannian manifold structure of the geometry. i challenge you to find an elementary proof in ordinary elementary geometry language.euclidean and hyperbolic geometry are the first examples of riemannian manifolds, and it is of great interest and not at all trivial to consider the differentiability of the dependences in geometry. e.g. the fact that SAS implies congruence means that the length of the third side of a triangle is a function of the lengths of the first two, and the angle. one can ask if this dependence is differentiable, and in particular how to compute the partial deriavtives of this dependence.
this idea is the basis for the proof i have been shown by robert foote of wabash college.
i do not know how to answer the question posed in this thread of interest in geometry. usually the problem of low interest in a subject stems from a bad introduction in a bad course. the material itself is always fascinating if exhibited by someone who knows and loves it.
i used to hate real analysis, but when i hear an expert analyst, and an expert instructor, speak about it i am usually entranced and instructed.
i also used to dislike the algebraic formulation of geometric ideas in abstract algebraic geometry, but after years of patient instruction from robert varley, and reading books by mumford and hartshorne, i find it beautiful and deep.
a first instance of this interaction is the field theory formualtion of constructibility, which builds on the insight of descartes, as hartshorne makes clear, and uses, not galois theory, but merely the dimension theory of field extensions, as one of the early directors of MSRI originally pointed out to me. (A senior moment hides his name, a famous ring theorist.)...Irving Kaplansky.
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