What is Euclidean Geometry
II. For a long time the question whether the parallel postulate can be proven on the basis of the other axioms and postulates has been controversial. This so-called parallel problem occupied generations of mathematicians and resulted in countless attempts at proof< agenum="" _86="" hervor,="" die="" jedoch="" alle="" daran="" scheiterten,="" daß="" unbemerkt="" aussagen="" verwendet="" wurden,="" die="" nicht="" aus="" den="" übrigen="" axiomen="" und="" postulaten="" ableitbar="" sind="" und="" vielmehr="" zum="" 5.="" postulat="" äquivalente="" aussagen="">
It was only when Gauss, Bolyai and Lobachevsky were able to show in the first half of the 19th century that the theory, which consists of the other axioms and postulates as well as the negation of the 5th postulate, is a consistent theory (namely a non-Euclidean geometry), the problem was solved, albeit in a completely unexpected way.
Since then, Euclidean geometry in the sense of an axiomatic structure of geometry has been the geometry in which all axioms of geometry, i.e. both those of absolute geometry and the 5th postulate of Euclid or the Euclidean axiom of parallels apply.
III. The Erlangen program, which Felix Klein presented in his inaugural lecture in 1872, represents a completely different way to the description of different geometries than the axiomatic way. According to this, a geometry can be understood as an invariant theory with respect to a transformation group on a set. The (point) set on which the Euclidean geometry is based is the Euclidean point space: an affine point space with an associated vector space on which a positively definite symmetrical bilinear form (scalar product) is defined. The transformation group, which characterizes the Euclidean geometry, is the group of movements (Euclidean movement group), the corresponding transformation group of the Euclidean vector space is the group of the orthogonal mappings, i. H. those mappings that leave the scalar product of two vectors unchanged. Thus, movements are equidistant images of the Euclidean point space on themselves.
Euclidean geometry in the sense of the Erlangen program is thus the theory of invariants with regard to the movements of a Euclidean point space. These invariants include, among others. Line lengths, line ratios, areas or volumes and the dimensions of angles, whereby the invariance of line lengths distinguishes Euclidean geometry from other (more general) geometries.
Compare this with Felix Klein's Erlangen program.
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