Why Euclidean Geometry?

e of the things evey normal human child learns in infancy, and this learning appears to be part of our biological programming. It is true that modern physical theories tell us that it is not strictly true; objects actually change shape as they move. But we are programmed not to think about these changes. This would seem to commit us to one of three geometries: euclidean, hyperbolic, and elliptical. The second technical result is John Wallis' proof that, in the presence of the axioms common to euclidean and non-euclidean geometry, the existence of triangles that are similar but not congruent is equivalent to the parallel postulate. (See Lanczos [7, p. 59].) The significance of this is that we are all used to thinking in terms of exact scale models, and this is true of every human culture of which I have ever heard. 1 But scale models require figures that are the same shape but di#erent sizes, and are thus possible only if there exist triangles which are simila