What is Euclidean geometry?
'''Euclidean geometry''' is a [[system]] in [[mathematics]]. People think [[Euclid]] was the first person who described it; therefore, it bears his name. He first described it in his textbook ''[[Euclid's Elements|Elements]]''. The book was the first systematic discussion of [[geometry]] as it was known at the time. In the book, Euclid first assumes a few [[axiom]]s. These form the base for later work. They are intuitively clear. Starting from those axioms, other [[theorem]]s can be [[wikt:proof|proven]]. In the [[19th century]], other forms of geometry were discovered. These are usually dubbed [[non-Euclidean geometry|non-Euclidean geometries]]. [[Carl Friedrich Gauss]], [[János Bolyai]], and [[Nikolai Ivanovich Lobachevsky]] were a few people that (independently of eachother) developed such geometries. Very often, these do not use the [[parallel (geometry)|parallel]] [[postulate]], but the other four axioms. ==The axioms== Euclid makes the following assumptions. These are axioms, and need not be proved. #Any two [[point (geometry)|point]]s can be joined by a straight [[line]]. #Any straight line segment can be made longer (extended) to infinity, so it becomes a straight line. #With a straight line segment it is possible to draw a circle, so that one endpoint of the segment is the center of the circle, and the other endpoint lies on the circle. The line segment becomes the radius of the circle. #All [[right angle]]s are [[congruence|congruent]]. # [[Parallel postulate]]. If two lines intersect a third in such a way that the sum of the inner [[angle]]s 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. ==Status== Euclidean geometry is a [[first-order theory]]. With it, statements like ''For all [[triangle]]s...'' can be made, and be proven. Statements like ''For all [[set]]s of triangles...'' are outside the [[scope]] of the theory. {{Mathematical logic}} {{authority control}} [[Category:Fields of geometry]]