Pasch axiom
One of the order axioms in the Hilbert system of axioms of Euclidean geometry. The statement of the axiom uses the concept "lies within (between) a segment" , and a segment is regarded here as a system of two distinct points and
; points lying "between"
and
are said to be points (or interior points) of the segment. The concept "between" (lying between) is described by the group of axioms of order, to which Pasch' axiom also belongs. It can be stated as follows: Let
,
and
be three non-collinear points and let
be a straight line in the plane
of these three points that does not pass through any of them; if the line passes through a point between
and
, then it must also pass through a point between
and
or a point between
and
.
This is an axiom of absolute geometry. By means of Hilbert's other axioms of order it can be proved that the line cannot intersect both
and
. M. Pasch stated this axiom in [1].
References
[1] | M. Pasch, "Vorlesungen über neuere Geometrie" , Springer, reprint (1926) |
[2] | D. Hilbert, "Grundlagen der Geometrie" , Teubner, reprint (1962) |
Comments
Sometimes the Veblen–Young axiom of projective geometry is, erroneously, called Pasch' axiom.
References
[a1] | H.S.M. Coxeter, "Introduction to geometry" , Wiley (1969) pp. 178 |
Pasch axiom. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pasch_axiom&oldid=13442