# Euclidean geometry

The geometry of space described by the system of axioms first stated systematically (though not sufficiently rigorous) in the *Elements* of Euclid. The space of Euclidean geometry is usually described as a set of objects of three kinds, called "points" , "lines" and "planes" ; the relations between them are incidence, order ( "lying between" ), congruence (or the concept of a motion), and continuity. The parallel axiom (fifth postulate) occupies a special place in the axiomatics of Euclidean geometry. The first sufficiently precise axiomatization of Euclidean geometry was given by D. Hilbert (see Hilbert system of axioms). There are modifications of Hilbert's axiom system as well as other versions of the axiomatics of Euclidean geometry. For example, in the vector axiomatics the concept of a vector is taken as one of the basic concepts. On the other hand the relation of symmetry may be taken as a basis for the axiomatics of plane Euclidean geometry (see [5]).

#### References

[1] | D. Hilbert, "Grundlagen der Geometrie" , Springer (1913) |

[2] | V.F. Kagan, "Foundations of geometry" , 1 , Moscow-Leningrad (1949) (In Russian) |

[3] | A.V. Pogorelov, "Foundations of geometry" , Moscow (1968) (In Russian) |

[4] | P.S. Alexandroff [P.S. Aleksandrov] (ed.) et al. (ed.) , Enzyklopaedie der Elementarmathematik , 4. Geometrie , Deutsch. Verlag Wissenschaft. (1969) (Translated from Russian) |

[5] | F. Bachmann, "Aufbau der Geometrie aus dem Spiegelungbegriff" , Springer (1973) |

#### Comments

A translation of Hilbert's monograph [1] is [a1].

#### References

[a1] | D. Hilbert, "Foundations of geometry" , Open Court (1971) (Translated from German) |

[a2] | M. Berger, "Geometry" , I , Springer (1987) |

[a3] | M. Greenberg, "Euclidean and non-Euclidean geometry" , Freeman (1980) |

[a4] | H. Busemann, "Recent synthetic geometry" , Springer (1970) |

[a5] | G. Choquet, "Geometry in a modern setting" , Kershaw (1969) |

**How to Cite This Entry:**

Euclidean geometry.

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Euclidean_geometry&oldid=34034