# Euclidean axiom by Hilbert

In Hilbert’s Grundlagen der Geometrie (‘Foundations of Geometry’; the original edition in 1899) there is the following argumentation.

Let $\alpha$ be an arbitrary plane, $a$ a line in $\alpha$ and $A$ a point in $\alpha$ which lies outside $a$. If we draw in $\alpha$ a line $c$ which passes through $A$ and intersects $a$ and then through $A$ a line $b$ such that the line $c$ intersects the lines $a$, $b$ with equal alternate interior angles (“unter gleichen Gegenwinkeln”), then it follows easily from the theorem on the outer angles, that the lines $a$, $b$ have no common point, i.e., in a plane $\alpha$ one can always draw otside a line $a$ another line which does not intersect the line $a$.

The Parallel Axiom reads now:

IV (). Let $a$ be an arbitrary line and $A$ be a point outside $a$: then in the plane determined by $a$ and $A$ there exists at most one line which passes through $A$ and does not intersect $a$.

Explanation. According the the preceding text and on grounds of the Parallel Axiom we realize, that there is one and only one line which passes through $A$ and do not intersect $a$; that is called the parallel   of $a$ through $A$.

The Parallel Axiom means the same as the following requirement:

When two lines $a$, $b$ in a plane do not meet a third line $c$ of the same plane, then also they do not meet each other.

The theorem on the outer angles is the following: An outer angle of a triangle  is greater than both non-adjacent angles of the triangle. Using this one may indirectly justify the assertion in the first cited paragraph.

If we , then we obtain easily the following well-known fact:

Theorem 31. http://planetmath.org/GeometricCongruence) of the corresponding or alternate interior angles implies that the lines are parallel.

## References

• 1 D. Hilbert: Grundlagen der Geometrie. Neunte Auflage, revidiert und ergänzt von Paul Bernays.  B. G. Teubner Verlagsgesellschaft, Stuttgart (1962).
Title Euclidean axiom by Hilbert EuclideanAxiomByHilbert 2013-03-22 17:19:16 2013-03-22 17:19:16 pahio (2872) pahio (2872) 9 pahio (2872) Topic msc 51M05 msc 51-01 CorrespondingAnglesInTransversalCutting AxiomaticGeometry