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 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 axioms. These form the base for later work. They are intuitively clear. Starting from those axioms, other theorems can be proven.

In the 19th century, other forms of geometry were discovered. These are usually dubbed 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 postulate, but the other four axioms.

The axioms

Euclid makes the following assumptions. These are axioms, and need not be proved.

1. Any two points can be joined by a straight line.
2. Any straight line segment can be made longer (extended) to infinity, so it becomes a straight line.
3. 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.
4. All right angles are congruent.
5. Parallel postulate. 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.

Status

Euclidean geometry is a first-order theory. With it, statements like For all triangles... can be made, and be proven. Statements like For all sets of triangles... are outside the scope of the theory.