Editing Erlangen program (section)

==The problems of nineteenth century geometry==

Since [[Euclid]], geometry had meant the geometry of [[Euclidean space]] of two dimensions ([[Euclidean plane geometry|plane geometry]]) or of three dimensions ([[solid geometry]]). In the first half of the nineteenth century there had been several developments complicating the picture. Mathematical applications required geometry of [[higher dimensions|four or more dimensions]]; the close scrutiny of the foundations of the traditional Euclidean geometry had revealed the independence of the [[parallel postulate]] from the others, and [[non-Euclidean geometry]] had been born. Klein proposed an idea that all these new geometries are just special cases of the [[projective geometry]], as already developed by [[Jean-Victor Poncelet|Poncelet]], [[August Ferdinand Möbius|Möbius]], [[Arthur Cayley|Cayley]] and others. Klein also strongly suggested to mathematical ''physicists'' that even a moderate cultivation of the projective purview might bring substantial benefits to them.

With every geometry, Klein associated an underlying [[symmetry group|group of symmetries]]. The hierarchy of geometries is thus mathematically represented as a hierarchy of these [[group (mathematics)|groups]], and hierarchy of their [[invariant (mathematics)|invariants]]. For example, lengths, angles and areas are preserved with respect to the [[Euclidean group]] of symmetries, while only the [[incidence structure]] and the [[cross-ratio]] are preserved under the most general [[projective geometry|projective transformations]]. A concept of [[parallel (geometry)|parallel]]ism, which is preserved in [[affine geometry]], is not meaningful in [[projective geometry]]. Then, by abstracting the underlying [[group (mathematics)|groups]] of symmetries from the geometries, the relationships between them can be re-established at the group level. Since the group of affine geometry is a [[subgroup]] of the group of projective geometry, any notion invariant in projective geometry is ''a priori'' meaningful in affine geometry; but not the other way round. If you remove required symmetries, you have a more powerful theory but fewer concepts and theorems (which will be deeper and more general).