Euclidean axiom by Hilbert
In Hilbert’s Grundlagen der Geometrie (‘Foundations of Geometry’; the original edition in 1899) there is the following argumentation.
Let be an arbitrary plane, a line in and a point in which lies outside . If we draw in a line which passes through and intersects and then through a line such that the line intersects the lines , with equal alternate interior angles (“unter gleichen Gegenwinkeln”), then it follows easily from the theorem on the outer angles, that the lines , have no common point, i.e., in a plane one can always draw otside a line another line which does not intersect the line .
The Parallel Axiom reads now:
IV (). Let be an arbitrary line and be a point outside : then in the plane determined by and there exists at most one line which passes through and does not intersect .
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 and do not intersect ; that is called the parallel of through .
The Parallel Axiom means the same as the following requirement:
When two lines , in a plane do not meet a third line 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. If two parallels intersect a third line, then the corresponding angles and the alternate interior angles are congruent, and conversely: the congruence (http://planetmath.org/GeometricCongruence) of the corresponding or alternate interior angles implies that the lines are parallel.
- 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|
|Date of creation||2013-03-22 17:19:16|
|Last modified on||2013-03-22 17:19:16|
|Last modified by||pahio (2872)|