Editing Euclidean space (section)

===History of the definition===
Euclidean space was introduced by [[Greek mathematics|ancient Greeks]] as an abstraction of our physical space. Their great innovation, appearing in [[Euclid's Elements|Euclid's ''Elements'']] was to build and ''[[proof (mathematics)|prove]]'' all geometry by starting from a few very basic properties, which are abstracted from the physical world, and cannot be mathematically proved because of the lack of more basic tools. These properties are called [[postulate]]s, or [[axiom]]s in modern language. This way of defining Euclidean space is still in use under the name of [[synthetic geometry]].

In 1637, [[René Descartes]] introduced [[Cartesian coordinates]], and showed that these allow reducing geometric problems to algebraic computations with numbers. This reduction of geometry to [[algebra]] was a major change in point of view, as, until then, the [[real number]]s were defined in terms of lengths and distances.

Euclidean geometry was not applied in spaces of dimension more than three until the 19th century. [[Ludwig Schläfli]] generalized Euclidean geometry to spaces of dimension {{mvar|n}}, using both synthetic and algebraic methods, and discovered all of the regular [[polytope]]s (higher-dimensional analogues of the [[Platonic solid]]s) that exist in Euclidean spaces of any dimension.{{sfn|Coxeter|1973|}}

Despite the wide use of Descartes' approach, which was called [[analytic geometry]], the definition of Euclidean space remained unchanged until the end of 19th century. The introduction of abstract [[vector space]]s allowed their use in defining Euclidean spaces with a purely algebraic definition. This new definition has been shown to be equivalent to the classical definition in terms of geometric axioms. It is this algebraic definition that is now most often used for introducing Euclidean spaces.