View source for Pasch axiom

One of the order axioms in the [[Hilbert system of axioms|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071800/p0718001.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071800/p0718002.png" />; points lying  "between"  <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071800/p0718003.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071800/p0718004.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071800/p0718005.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071800/p0718006.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071800/p0718007.png" /> be three non-collinear points and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071800/p0718008.png" /> be a straight line in the plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071800/p0718009.png" /> of these three points that does not pass through any of them; if the line passes through a point between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071800/p07180010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071800/p07180011.png" />, then it must also pass through a point between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071800/p07180012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071800/p07180013.png" /> or a point between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071800/p07180014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071800/p07180015.png" />.

This is an axiom of [[Absolute geometry|absolute geometry]]. By means of Hilbert's other axioms of order it can be proved that the line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071800/p07180016.png" /> cannot intersect both <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071800/p07180017.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071800/p07180018.png" />. M. Pasch stated this axiom in [[#References|[1]]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M. Pasch,   "Vorlesungen über neuere Geometrie" , Springer, reprint  (1926)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  D. Hilbert,   "Grundlagen der Geometrie" , Teubner, reprint  (1962)</TD></TR></table>



====Comments====
Sometimes the Veblen–Young axiom of [[Projective geometry|projective geometry]] is, erroneously, called Pasch' axiom.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H.S.M. Coxeter,   "Introduction to geometry" , Wiley  (1969)  pp. 178</TD></TR></table>