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408:(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.
222:
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.
261:. 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
281:
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
310:
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).
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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.
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There is no mention of internal and external intersections in the statement of the Veblen-Young axiom which is only concerned with the
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basis for
Euclidean geometry. Depending upon the edition, it is numbered either II.4 or II.5. His statement is given above.
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to our line, we count an "intersection at infinity" as external.) A more informal version of the axiom is often seen:
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If a line, not passing through any vertex of a triangle, meets one side of the triangle then it meets another side.
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In
Hilbert's treatment, this axiom appears in the section concerning axioms of order and is referred to as a
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It can, however, be derived from weaker axioms of plane separation taken for granted by Euclid, as shown in
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Pambuccian, Victor (2011), "The axiomatics of ordered geometry: I. Ordered incidence spaces.",
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If a line intersects two sides of a triangle, then it also intersects the third side.
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only
Hilbert's axioms I.1,2,3 and II.1,2,3 are needed for this. Proof is given in
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This is taken from the Unger translation of the 10th edition of
Hilbert's
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Pambuccian, Victor (2024), "Why did Euclid not need the Pasch axiom?.",
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Pasch's axiom should not be confused with the Veblen-Young axiom for
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Euclidean and Non-Euclidean
Geometries: Development and History
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Euclidean and Non-Euclidean
Geometries: Development and History
590:(Third ed.), Addison-Wesley, Reading, MA, p. 74,
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as Euclid gave them. Its essential role was discovered by
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Projective geometry: from foundations to applications
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Foundations of
Euclidean and Non-Euclidean Geometry
162:A more modern version of this axiom is as follows:
155:is proved in Supplement I,1, which was written by
647:(in German) (2nd ed.), Leipzig: B.G. Teubner
172:In the plane, if a line intersects one side of a
446:Beutelspacher, Albrecht; Rosenbaum, Ute (1998),
131:, it also passes through a point of the segment
588:Elementary Geometry from an Advanced Standpoint
519:(4th ed.), San Francisco: W.H. Freeman,
501:(1st ed.), San Francisco: W.H. Freeman,
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96:be three points that do not lie on a
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544:(in German), Leipzig: B.G. Teubner
431:Beutelspacher & Rosenbaum 1998
378:Beutelspacher & Rosenbaum 1998
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644:Vorlesungen uber neuere Geometrie
480:, New York: Marcel Dekker, Inc.,
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265:with the sides of the triangle
246:uses Pasch's axiom in his book
515:Greenberg, Marvin Jay (2007),
239:Hilbert's use of Pasch's axiom
1:
615:10.1016/j.exmath.2010.09.004
63:and meeting the other sides
552:The Foundations of Geometry
191:(In case the third side is
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670:Wylie, Jr., C.R. (2009) ,
632:10.1007/s00022-024-00712-x
476:Faber, Richard L. (1983),
452:Cambridge University Press
292:, which may be stated as:
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656:, New York: McGraw-Hill,
606:Expositiones Mathematicae
726:Euclidean plane geometry
563:Hilbert, David (1999) ,
549:Hilbert, David (1950) ,
541:Grundlagen der Geometrie
404:axiom II.5 in Hilbert's
16:Not to be confused with
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35:, used implicitly by
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624:Journal of Geometry
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45:Moritz Pasch
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419:Faber (1983
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720:Categories
440:References
341:Pasch 1912
285:, p. 67).
185:externally
181:internally
177:internally
157:P. Bernays
69:externally
65:internally
61:internally
41:postulates
707:MathWorld
641:(1912) ,
252:axiomatic
51:Statement
47:in 1882.
586:(1990),
538:(1903),
497:(1974),
193:parallel
174:triangle
100:and let
25:geometry
626:(115),
470:1629468
273:Caveats
225:Hilbert
210:History
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37:Euclid
556:(PDF)
314:Notes
106:plane
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482:ISBN
456:ISBN
147:and
98:line
84:Let
67:and
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267:ABC
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109:ABC
23:In
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466:MR
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397:^
333:^
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27:,
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630::
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121:C
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86:A
20:.
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