Pasch's axiom

In geometry, Pasch's axiom is a statement in plane geometry, used implicitly by Euclid, which cannot be derived from Euclid's postulates. Its essential role was discovered by Moritz Pasch.

Contents 1 Statement 2 History 3 Equivalences 4 Caveats 5 Notes 6 References 7 External links

Statement

The axiom states that, in the plane,^[1]

If a line intersects one side of a triangle internally then it intersects precisely one other side internally and the third side externally, if it does not pass through a vertex of the triangle.

A more informal version of the axiom is often seen:

If a line, not passing through any vertex of a triangle, meets one side of the triangle then it meets another side.

In this statement, a side of a triangle is taken to be the line segment joining two vertices of the triangle, so the expression "a line meeting a side" means that the line meets the line forming the side of the triangle internally.

History

Pasch published this axiom in 1882,^[2] and showed that Euclid's axioms were incomplete. The axiom was part of Pasch's approach to introducing the concept of order into plane geometry.

Equivalences

In other treatments of elementary geometry, using different sets of axioms, Pasch's axiom can be proved as a theorem;^[3] it is a consequence of the plane separation axiom when that is taken as one of the axioms. Hilbert uses Pasch's axiom in his axiomatic treatment of Euclidean geometry.^[4] Given the remaining axioms in Hilbert's system, it can be shown that Pasch's axiom is logically equivalent to the plane separation axiom.^[5]

Caveats

Pasch's axiom is distinct from Pasch's theorem which is a statement about the order of points on a line.

Pasch's axiom should not be confused with the Veblen-Young axiom^[6], which may be stated as:

If a line intersects two sides of a triangle, then it also intersects the third side.

Notice that there is no mention of internal and external intersections in this axiom which is only concerned with the lines meeting (incidence).

Notes

References

• Beutelspacher, Albrecht; Rosenbaum, Ute (1998), Projective geometry: from foundations to applications, Cambridge University Press, ISBN 978-0-521-48277-6; 978-0-521-48364-3, MR1629468 
• Faber, Richard L. (1983), Foundations of Euclidean and Non-Euclidean Geometry, New York: Marcel Dekker, Inc., ISBN 0-8247-1748-1 
• Edwin Moise. Elementary Geometry from an Advanced Standpoint, Third Edition. Addison-Wesley, Reading, MA, 1990. Page 74.
• Victor Pambuccian: The axiomatics of ordered geometry: I. Ordered incidence spaces. Expositiones Mathematicae 29 (2011), 24-66.
• Wylie,Jr., C.R. (1964), Foundations of Geometry, New York: McGraw-Hill 

External links

• Pasch's Axiom MathWorld