Editing Euclidean geometry (section)

===Axioms===
[[File:Parallel postulate en.svg|thumb|The parallel postulate (Postulate 5): If two lines intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough.]]

Euclidean geometry is an [[axiomatic system]], in which all [[theorem]]s ("true statements") are derived from a small number of simple axioms. Until the advent of [[non-Euclidean geometry]], these axioms were considered to be obviously true in the physical world, so that all the theorems would be equally true. However, Euclid's reasoning from assumptions to conclusions remains valid independently from the physical reality.<ref name=Wolfe>The assumptions of Euclid are discussed from a modern perspective in
{{cite book |title=Introduction to Non-Euclidean Geometry |author=Harold E. Wolfe |url=https://books.google.com/books?id=VPHn3MutWhQC&pg=PA9 |page=9 |isbn=978-1-4067-1852-2 |year=2007 |publisher=Mill Press}}
</ref>

Near the beginning of the first book of the ''Elements'', Euclid gives five [[postulate]]s (axioms) for plane geometry, stated in terms of constructions (as translated by Thomas Heath):<ref>tr. Heath, pp. 195–202.</ref>

:Let the following be postulated:
# To draw a [[straight line]] from any [[Point (geometry)|point]] to any point.
# To produce (extend) a [[Line segment|finite straight line]] continuously in a straight line.
# To describe a [[circle]] with any centre and distance (radius).
# That all [[right angle]]s are equal to one another.
# [The [[parallel postulate]]]: That, if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which the angles are less than two right angles.

Although Euclid explicitly only asserts the existence of the constructed objects, in his reasoning he also implicitly assumes them to be unique.

The ''Elements'' also include the following five "{{Visible anchor|common notions}}":

# Things that are equal to the same thing are also equal to one another (the [[transitive property]] of a [[Euclidean relation]]).
# If equals are added to equals, then the wholes are equal (Addition property of equality).
# If equals are subtracted from equals, then the differences are equal (subtraction property of equality).
# Things that coincide with one another are equal to one another (reflexive property).
# The whole is greater than the part.

Modern scholars agree that Euclid's postulates do not provide the complete logical foundation that Euclid required for his presentation.<ref>{{citation|first=Gerard A.|last=Venema|title=Foundations of Geometry|year=2006|publisher=Prentice-Hall|page=8|isbn=978-0-13-143700-5}}.</ref> Modern [[Foundations of geometry|treatments]] use more extensive and complete sets of axioms.