Pasch's theorem
{{short description|Result about 4 points on a line which cannot be derived from Euclid's postulates}}
{{distinguish|text=Pasch's axiom regarding a line through a triangle}}
In geometry, Pasch's theorem, stated in 1882 by the German mathematician Moritz Pasch,{{Harvnb|Pasch|1912}} is a result in plane geometry which cannot be derived from Euclid's postulates.
Statement
The statement is as follows: {{math_theorem|name=Pasch's theorem|Given points {{mvar|a}}, {{mvar|b}}, {{mvar|c}}, and {{mvar|d}} on a line, if it is known that the points are ordered as ({{mvar|a}}, {{mvar|b}}, {{mvar|c}}) and ({{mvar|b}}, {{mvar|c}}, {{mvar|d}}), then it is also true that ({{mvar|a}}, {{mvar|b}}, {{mvar|d}}).{{harvtxt|Coxeter|1969|p=179}} states the result in 12.274 but does not refer to it specifically as Pasch's theorem.}} [Here, for example, ({{mvar|a}}, {{mvar|b}}, {{mvar|c}}) means that point {{mvar|b}} lies between points {{mvar|a}} and {{mvar|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,{{citation|first=E. H.|last=Moore|title=On the projective axioms of geometry|journal=Transactions of the American Mathematical Society|year=1902|volume=3|issue=1 |pages=142–158|url=https://www.ams.org/journals/tran/1902-003-01/S0002-9947-1902-1500592-8/S0002-9947-1902-1500592-8.pdf|doi=10.2307/1986321|jstor=1986321 |doi-access=free}} 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.
See also
Notes
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References
- {{citation | last=Coxeter | first=H.S.M. | authorlink=Harold Scott MacDonald Coxeter | title=Introduction to geometry | edition=2nd | publisher=John Wiley and Sons | year=1969 | isbn=978-0-471-18283-2 | zbl=0181.48101 | url-access=registration | url=https://archive.org/details/introductiontoge0002coxe }}
- {{citation|first=Moritz|last=Pasch|author-link=Moritz Pasch|title=Vorlesungen uber neuere Geometrie|year=1912|language=de|orig-year=first edition 1882|url=http://quod.lib.umich.edu/cgi/t/text/text-idx?c=umhistmath;idno=ABV7607 |place=Leipzig|publisher=B.G. Teubner|edition=2nd}}
External links
- {{MathWorld|title=Pasch's Theorem|urlname=PaschsTheorem}}
Category:Euclidean plane geometry
Category:Foundations of geometry
Category:Theorems in plane geometry
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