Editing Euclidean geometry (section)

===Methods of proof===
Euclidean Geometry is ''[[Constructive proof|constructive]]''. Postulates 1, 2, 3, and 5 assert the existence and uniqueness of certain geometric figures, and these assertions are of a constructive nature: that is, we are not only told that certain things exist, but are also given methods for creating them with no more than a [[compass and straightedge|compass and an unmarked straightedge]].<ref>Ball, p.&nbsp;56.</ref> In this sense, Euclidean geometry is more concrete than many modern axiomatic systems such as [[set theory]], which often assert the existence of objects without saying how to construct them, or even assert the existence of objects that cannot be constructed within the theory.<ref name=set_theory>Within Euclid's assumptions, it is quite easy to give a formula for area of triangles and squares. However, in a more general context like set theory, it is not as easy to prove that the area of a square is the sum of areas of its pieces, for example. See [[Lebesgue measure]] and [[Banach–Tarski paradox]].</ref> Strictly speaking, the lines on paper are ''[[Scientific modelling|models]]'' of the objects defined within the formal system, rather than instances of those objects. For example, a Euclidean straight line has no width, but any real drawn line will have. Though nearly all modern mathematicians consider [[Non-constructive proof|nonconstructive proofs]] just as sound as constructive ones, they are often considered less [[Mathematical beauty|elegant]], intuitive, or practically useful. Euclid's constructive proofs often supplanted fallacious nonconstructive ones, e.g. some Pythagorean proofs that assumed all numbers are rational, usually requiring a statement such as "Find the greatest common measure of ..."<ref>{{cite book|author=Daniel Shanks|title=Solved and Unsolved Problems in Number Theory|year=2002|publisher=American Mathematical Society}}</ref>

Euclid often used [[proof by contradiction]].<ref>{{cite journal |title=On the Status of Proofs by Contradiction in the Seventeenth Century |first=Paolo |last=Mancosu |journal=Synthese |year=1991 |volume=88 |number=1 |pages=15–41 |doi=10.1007/BF00540091 |jstor=20116923 }}</ref>