Editing Synthetic geometry (section)

{{Short description|Geometry without using coordinates}}
{{General geometry |branches}}

'''Synthetic geometry''' (sometimes referred to as '''axiomatic geometry''' or even '''pure geometry''') is  [[geometry]] without the use of [[coordinates]]. It relies on the [[axiomatic method]] for proving all results from a few basic properties initially called [[postulate]]s, and at present called [[axiom]]s. 

After the 17th-century introduction by [[René Descartes]] of the coordinate method, which was called [[analytic geometry]], the term "synthetic geometry" was coined to refer to the older methods that were, before Descartes, the only known ones.

According to [[Felix Klein]]
<blockquote>
Synthetic geometry is that which studies [[shape|figures]] as such, without recourse to formulae, whereas analytic geometry consistently makes use of such formulae as can be written down after the adoption of an appropriate system of coordinates.<ref>{{harvnb|Klein|1948|loc=p. 55}}</ref>
</blockquote>

The first systematic approach for synthetic geometry is [[Euclid's Elements|Euclid's ''Elements'']].  However, it appeared at the end of the 19th century that [[Euclid]]'s postulates were not sufficient for characterizing geometry. The first complete [[axiom system]] for geometry was given only at the end of the 19th century by [[David Hilbert]]. At the same time, it appeared that both synthetic methods and analytic methods can be used to build geometry. The fact that the two approaches are equivalent has been proved by [[Emil Artin]] in his book ''[[Geometric Algebra (book)|Geometric Algebra]]''.

Because of this equivalence, the distinction between synthetic and analytic geometry is no more in use, except at elementary level, or for geometries that are not related to any sort of numbers, such as some [[finite geometry|finite geometries]] and [[non-Desarguesian geometry]].{{cn|date=February 2023}}