In geometry, Pasch's axiom is a statement in plane geometry, used implicitly by Euclid, which cannot be derived from the postulates as Euclid gave them. Its essential role was discovered by Moritz Pasch in 1882.
Statement
The axiom states that,
Pasch's axiom — Let A, B, C be three points that do not lie on a line and let a be a line in the plane ABC which does not meet any of the points A, B, C. If the line a passes through a point of the segment AB, it also passes through a point of the segment AC, or through a point of segment BC.
The fact that segments AC and BC are not both intersected by the line a is proved in Supplement I,1, which was written by P. Bernays.
A more modern version of this axiom is as follows:
A more modern version of Pasch's axiom — In the plane, 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.
(In case the third side is parallel to our line, we count an "intersection at infinity" as external.) A more informal version of the axiom is often seen:
A more informal version of Pasch's axiom — If a line, not passing through any vertex of a triangle, meets one side of the triangle then it meets another side.
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 uses Pasch's axiom in his axiomatic treatment of Euclidean geometry. 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 a with the sides of the triangle ABC.
Caveats
Pasch's axiom is distinct from Pasch's theorem 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 Greenberg (1974, p. 67).
Pasch's axiom should not be confused with the Veblen-Young axiom for projective geometry, which may be stated as:
Veblen-Young axiom for projective geometry — If a line intersects two sides of a triangle, then it also intersects the third side.
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
- It can, however, be derived from weaker axioms of plane separation taken for granted by Euclid, as shown in Pambuccian 2024
- Pasch 1912, p. 21
- This is taken from the Unger translation of the 10th edition of Hilbert's Foundations of Geometry and is numbered II.4.
- Hilbert 1999, p. 200, the Unger translation.
- Beutelspacher & Rosenbaum 1998, p. 7
- Wylie, Jr. 1964, p. 100
- 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.
- only Hilbert's axioms I.1,2,3 and II.1,2,3 are needed for this. Proof is given in Faber (1983, pp. 116–117).
- Beutelspacher & Rosenbaum 1998, p. 6
References
- Beutelspacher, Albrecht; Rosenbaum, Ute (1998), Projective geometry: from foundations to applications, Cambridge University Press, ISBN 978-0-521-48364-3, MR 1629468
- Faber, Richard L. (1983), Foundations of Euclidean and Non-Euclidean Geometry, New York: Marcel Dekker, Inc., ISBN 978-0-8247-1748-3
- Greenberg, Marvin Jay (1974), Euclidean and Non-Euclidean Geometries: Development and History (1st ed.), San Francisco: W.H. Freeman, ISBN 978-0-7167-0454-6
- Greenberg, Marvin Jay (2007), Euclidean and Non-Euclidean Geometries: Development and History (4th ed.), San Francisco: W.H. Freeman, ISBN 978-0-7167-9948-1
- Hilbert, David (1903), Grundlagen der Geometrie (in German), Leipzig: B.G. Teubner
- Hilbert, David (1950) [1902], The Foundations of Geometry (PDF), translated by Townsend, E. J., LaSalle, IL: Open Court Publishing
- Hilbert, David (1999) [1971], Foundations of Geometry, translated by Unger, Leo (2nd ed.), LaSalle, IL: Open Court Publishing, ISBN 978-0-87548-164-7
- Moise, Edwin (1990), Elementary Geometry from an Advanced Standpoint (Third ed.), Addison-Wesley, Reading, MA, p. 74, ISBN 978-0-201-50867-3
- Pambuccian, Victor (2011), "The axiomatics of ordered geometry: I. Ordered incidence spaces.", Expositiones Mathematicae (29): 24–66, doi:10.1016/j.exmath.2010.09.004
- Pambuccian, Victor (2024), "Why did Euclid not need the Pasch axiom?.", Journal of Geometry (115), doi:10.1007/s00022-024-00712-x
- Pasch, Moritz (1912) [first edition 1882], Vorlesungen uber neuere Geometrie (in German) (2nd ed.), Leipzig: B.G. Teubner
- Wylie, Jr., Clarence Raymond (1964), Foundations of Geometry, New York: McGraw-Hill, ISBN 978-0-070-72191-3
- Wylie, Jr., C.R. (2009) [1964], Foundations of Geometry, Mineola, New York: Dover Publications, ISBN 978-0-486-47214-0
External links
- Weisstein, Eric W. "Pasch's Axiom". MathWorld.
In geometry Pasch s axiom is a statement in plane geometry used implicitly by Euclid which cannot be derived from the postulates as Euclid gave them Its essential role was discovered by Moritz Pasch in 1882 StatementTwo lines in black meeting a triangle side internally and meeting the other sides internally and externally The axiom states that Pasch s axiom Let A B C be three points that do not lie on a line and let a be a line in the plane ABC which does not meet any of the points A B C If the line a passes through a point of the segment AB it also passes through a point of the segment AC or through a point of segment BC The fact that segments AC and BC are not both intersected by the line a is proved in Supplement I 1 which was written by P Bernays A more modern version of this axiom is as follows A more modern version of Pasch s axiom In the plane 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 In case the third side is parallel to our line we count an intersection at infinity as external A more informal version of the axiom is often seen A more informal version of Pasch s axiom If a line not passing through any vertex of a triangle meets one side of the triangle then it meets another side HistoryPasch 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 EquivalencesIn 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 uses Pasch s axiom in his axiomatic treatment of Euclidean geometry 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 axiomDavid 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 a with the sides of the triangle ABC CaveatsPasch s axiom is distinct from Pasch s theorem 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 Greenberg 1974 p 67 Pasch s axiom should not be confused with the Veblen Young axiom for projective geometry which may be stated as Veblen Young axiom for projective geometry If a line intersects two sides of a triangle then it also intersects the third side 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 NotesIt can however be derived from weaker axioms of plane separation taken for granted by Euclid as shown in Pambuccian 2024 Pasch 1912 p 21 This is taken from the Unger translation of the 10th edition of Hilbert s Foundations of Geometry and is numbered II 4 Hilbert 1999 p 200 the Unger translation Beutelspacher amp Rosenbaum 1998 p 7 Wylie Jr 1964 p 100 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 only Hilbert s axioms I 1 2 3 and II 1 2 3 are needed for this Proof is given in Faber 1983 pp 116 117 Beutelspacher amp Rosenbaum 1998 p 6ReferencesBeutelspacher Albrecht Rosenbaum Ute 1998 Projective geometry from foundations to applications Cambridge University Press ISBN 978 0 521 48364 3 MR 1629468 Faber Richard L 1983 Foundations of Euclidean and Non Euclidean Geometry New York Marcel Dekker Inc ISBN 978 0 8247 1748 3 Greenberg Marvin Jay 1974 Euclidean and Non Euclidean Geometries Development and History 1st ed San Francisco W H Freeman ISBN 978 0 7167 0454 6 Greenberg Marvin Jay 2007 Euclidean and Non Euclidean Geometries Development and History 4th ed San Francisco W H Freeman ISBN 978 0 7167 9948 1 Hilbert David 1903 Grundlagen der Geometrie in German Leipzig B G Teubner Hilbert David 1950 1902 The Foundations of Geometry PDF translated by Townsend E J LaSalle IL Open Court Publishing Hilbert David 1999 1971 Foundations of Geometry translated by Unger Leo 2nd ed LaSalle IL Open Court Publishing ISBN 978 0 87548 164 7 Moise Edwin 1990 Elementary Geometry from an Advanced Standpoint Third ed Addison Wesley Reading MA p 74 ISBN 978 0 201 50867 3 Pambuccian Victor 2011 The axiomatics of ordered geometry I Ordered incidence spaces Expositiones Mathematicae 29 24 66 doi 10 1016 j exmath 2010 09 004 Pambuccian Victor 2024 Why did Euclid not need the Pasch axiom Journal of Geometry 115 doi 10 1007 s00022 024 00712 x Pasch Moritz 1912 first edition 1882 Vorlesungen uber neuere Geometrie in German 2nd ed Leipzig B G Teubner Wylie Jr Clarence Raymond 1964 Foundations of Geometry New York McGraw Hill ISBN 978 0 070 72191 3 Wylie Jr C R 2009 1964 Foundations of Geometry Mineola New York Dover Publications ISBN 978 0 486 47214 0External linksWeisstein Eric W Pasch s Axiom MathWorld
