Axiom of Pasch
The set of Pasch (by Moritz Pasch ) is in the synthetic geometry commonly called axiom used:
"Let A, B, C three not in a straight line located points and a one straight line in the plane meets ABC that none of these three points. If the straight line a then passes through a point on the segment AB, it certainly also goes either through a point on the segment BC or through a point on the segment AC. "
This can be clearly expressed as follows: “ If a straight line enters a triangle through one side, it certainly also emerges again through one side of the triangle. "
Pasch formulated this axiom in 1882. Euclid was not yet interested in the necessity of such an axiom. Evidence of this kind was used by him (and his disciples for the next 2,000 years) as a matter of course.
The formulation of this axiom therefore represents an important step on the way of geometry to a strictly axiomatic theory ( axiomatization ). It is one of the axioms by which a weak interrelation can be characterized on an affine level . In Hilbert's system of axioms of Euclidean geometry , it is one of the axioms that describe a (strong) interrelation and thus an arrangement of the plane.
→ Veblen-Young's axiom has also been referred to as Pasch's axiom in the mathematical literature.
literature
- Jeremy Gray : Worlds out of nothing: a course of the history of geometry of the 19th Century , Springer 2007
- Moritz Pasch: Lectures on newer geometry , Leipzig 1882
- Victor Pambuccian: The axiomatics of ordered geometry: I. Ordered incidence spaces. Expositiones Mathematicae 29 (2011), 24–66, doi : 10.1016 / j.exmath.2010.09.004 .