This system is an axiomatic system, which hoped to prove all the "true statements" as theorems in geometry from a set of finite number of axioms.

The five postulates/axioms of the Euclidean system are:

- Any two points can be joined by a straight line.
- Any straight line segment can be extended indefinitely in a straight line.
- Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center.
- All right angles are congruent.
- If two lines are drawn which 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. This postulate is equivalent to what is known as the
**parallel postulate**.

*Through a point not on a given straight line, one and only one line can be drawn that never meets the given line.*

Other mathematicians proposed that the statement was indeed a postulate, and tried to prove their suggestions, by "negating the postulate" (*call the negation NOT P*) as an axiom, hoping to arrive at a contradiction. If a contradiction is achieved with Y, a known theorem, *(previously deduced from the other postulates) ie*: if it leads to a situation - Y AND NOT Y, or if a contradiction is achieved directly with the assumed NOT P ie : if we arrive at P AND NOT P, this would mean that the assumption of negation of P was wrong (Proof by contradiction), and hence the parallel postulate needs to be "assumed to be true".

However, both factions of mathematicians were stumped in their efforts to achieve a definite answer to the question of "whether the parallel postulate is an axiom of geometry". The disbelievers, could not successfully prove that it is not. Neither could the believers arrive at a contradiction, by negating it. Curiously, however, by negating the fifth postulate in various ways, they extended geometry to represent non-planar universes. (See Non-Euclidean geometry for further explication)

In particular, this postulate separates Euclidean geometry from hyperbolic geometry, where many parallel lines could be drawn through the point, and from elliptic and projective geometry, where no parallel lines exist. (Euclidean geometry does, however, share the parallel postulate with some other geometries, such as certain finite geometries and affine geometry.)

Since Euclid's time, other mathematicians have laid out more thorough axiomatic systems for Euclidean geometry, such as David Hilbert and George Birkhoff.

Today Euclidean geometry is usually constructed rather than axiomatized, by means of analytic geometry. A rectangular coordinate system maps each point in Euclidean space with a unique list of n real numbers (x_{1},...,x_{n}), so we can define it to be the set of all such lists (**R**^{n}). We also define a metric (distance function) d by

See also: