Pasch's theorem

In geometry, Pasch's theorem, stated in 1882 by the German mathematician Moritz Pasch,^[1] is a result in plane geometry which cannot be derived from Euclid's postulates.

Statement

The statement is as follows:

Pasch's theorem — Given points $a$ , $b$ , $c$ , and $d$ on a line, if it is known that the points are ordered as ( $a$ , $b$ , $c$ ) and ( $b$ , $c$ , $d$ ), then it is also true that ( $a$ , $b$ , $d$ ).^[2]

[Here, for example, ( $a$ , $b$ , $c$ ) means that point $b$ lies between points $a$ and $c$ .]

Hilbert's use of Pasch's theorem

David Hilbert originally included Pasch's theorem as an axiom in his modern treatment of Euclidean geometry in The Foundations of Geometry (1899). However, it was found by E.H. Moore in 1902 that the axiom is redundant,^[3] and revised editions now list it as a theorem. Thus Pasch's theorem is also known as Hilbert's discarded axiom.

Pasch's axiom, a separate statement, is also included and remains an axiom in Hilbert's treatment.

Notes

^ Pasch 1912
^ Coxeter (1969, p. 179) states the result in 12.274 but does not refer to it specifically as Pasch's theorem.
^ Moore, E.H. (1902), "On the projective axioms of geometry" (PDF), Transactions of the American Mathematical Society, 3 (1): 142–158, doi:10.2307/1986321, JSTOR 1986321

References

Coxeter, H.S.M. (1969), Introduction to geometry (2nd ed.), John Wiley and Sons, ISBN 978-0-471-18283-2, Zbl 0181.48101
Pasch, Moritz (1912) [first edition 1882], Vorlesungen uber neuere Geometrie (in German) (2nd ed.), Leipzig: B.G. Teubner

External links

Weisstein, Eric W. "Pasch's Theorem". MathWorld.

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[1] Pasch 1912

[2] Coxeter (1969, p. 179) states the result in 12.274 but does not refer to it specifically as Pasch's theorem.

[3] Moore, E.H. (1902), "On the projective axioms of geometry" (PDF), Transactions of the American Mathematical Society, 3 (1): 142–158, doi:10.2307/1986321, JSTOR 1986321

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