geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...

, Pasch's axiom is a statement in

plane geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry: the '' Elements''. Euclid's approach consists in assuming a small set of intuitively appealing axioms ...

, used implicitly by

Euclid Euclid (; grc-gre, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the '' Elements'' treatise, which established the foundations of ...

, which cannot be derived from the

postulates An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or f ...

as Euclid gave them. Its essential role was discovered by

Moritz Pasch Moritz Pasch (8 November 1843, Breslau, Prussia (now Wrocław, Poland) – 20 September 1930, Bad Homburg, Germany) was a German mathematician of Jewish ancestry specializing in the foundations of geometry. He completed his Ph.D. at the Univer ...

in 1882.

Statement

The axiom states that, The fact that segments ''AC'' and ''BC'' are not both intersected by the line is proved in Supplement I,1, which was written by P. Bernays. A more modern version of this axiom is as follows: (In case the third side is

parallel Parallel is a geometric term of location which may refer to: Computing * Parallel algorithm * Parallel computing * Parallel metaheuristic * Parallel (software), a UNIX utility for running programs in parallel * Parallel Sysplex, a cluster of ...

to our line, we count an "intersection at infinity" as external.) A more informal version of the axiom is often seen:

History

Pasch published this axiom in 1882, 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; it is a consequence of the plane separation axiom when that is taken as one of the axioms.

Hilbert David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician, one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many ...

uses Pasch's axiom in his axiomatic treatment of

Euclidean geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry: the '' Elements''. Euclid's approach consists in assuming a small set of intuitively appealing axioms ...

.axiom II.5 in Hilbert's ''Foundations of Geometry'' (Townsend translation referenced below), in the authorized English translation of the 10th edition translated by L. Unger (also published by Open Court) it is numbered II.4. There are several differences between these translations. Given the remaining axioms in Hilbert's system, it can be shown that Pasch's axiom is logically equivalent to the plane separation axiom.

Hilbert's use of Pasch's axiom

David Hilbert uses Pasch's axiom in his book ''Foundations of Geometry'' which provides an axiomatic basis for Euclidean geometry. Depending upon the edition, it is numbered either II.4 or II.5. His statement is given above. In Hilbert's treatment, this axiom appears in the section concerning axioms of order and is referred to as a ''plane axiom of order''. Since he does not phrase the axiom in terms of the sides of a triangle (considered as lines rather than line segments) there is no need to talk about internal and external intersections of the line with the sides of the triangle ''ABC''.

Caveats

Pasch's axiom is distinct from

Pasch's theorem In geometry, Pasch's theorem, stated in 1882 by the German mathematician Moritz Pasch, is a result in plane geometry which cannot be derived from Euclid's postulates. Statement The statement is as follows: ere, for example, (, , ) means that ...

which is a statement about the order of four points on a line. However, in literature there are many instances where Pasch's axiom is referred to as Pasch's theorem. A notable instance of this is . Pasch's axiom should not be confused with the Veblen-Young axiom for

projective geometry In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting, ...

, which may be stated as: There is no mention of internal and external intersections in the statement of the Veblen-Young axiom which is only concerned with the incidence property of the lines meeting. In projective geometry the concept of betweeness (required to define internal and external) is not valid and all lines meet (so the issue of parallel lines does not arise).

Notes

References

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External links

*{{MathWorld, title=Pasch's Axiom, urlname=PaschsAxiom Euclidean plane geometry Foundations of geometry