Playfair's axiom

  • premise of playfair's axiom: a line and a point not on the line
    logical consequence of playfair's axiom: a second line, parallel to the first, passing through the point

    in geometry, playfair's axiom is an axiom that can be used instead of the fifth postulate of euclid (the parallel postulate):

    in a plane, given a line and a point not on it, at most one line parallel to the given line can be drawn through the point.[1]

    it is equivalent to euclid's parallel postulate in the context of euclidean geometry[2] and was named after the scottish mathematician john playfair. the "at most" clause is all that is needed since it can be proved from the remaining axioms that at least one parallel line exists. the statement is often written with the phrase, "there is one and only one parallel". in euclid's elements, two lines are said to be parallel if they never meet and other characterizations of parallel lines are not used.[3][4]

    this axiom is used not only in euclidean geometry but also in the broader study of affine geometry where the concept of parallelism is central. in the affine geometry setting, the stronger form of playfair's axiom (where "at most" is replaced by "one and only one") is needed since the axioms of neutral geometry are not present to provide a proof of existence. playfair's version of the axiom has become so popular that it is often referred to as euclid's parallel axiom,[5] even though it was not euclid's version of the axiom. a corollary of the axiom is that the binary relation of parallel lines is a serial relation.

  • history
  • relation with euclid's fifth postulate
  • transitivity of parallelism
  • notes
  • references

Premise of Playfair's axiom: a line and a point not on the line
Logical consequence of Playfair's axiom: a second line, parallel to the first, passing through the point

In geometry, Playfair's axiom is an axiom that can be used instead of the fifth postulate of Euclid (the parallel postulate):

In a plane, given a line and a point not on it, at most one line parallel to the given line can be drawn through the point.[1]

It is equivalent to Euclid's parallel postulate in the context of Euclidean geometry[2] and was named after the Scottish mathematician John Playfair. The "at most" clause is all that is needed since it can be proved from the remaining axioms that at least one parallel line exists. The statement is often written with the phrase, "there is one and only one parallel". In Euclid's Elements, two lines are said to be parallel if they never meet and other characterizations of parallel lines are not used.[3][4]

This axiom is used not only in Euclidean geometry but also in the broader study of affine geometry where the concept of parallelism is central. In the affine geometry setting, the stronger form of Playfair's axiom (where "at most" is replaced by "one and only one") is needed since the axioms of neutral geometry are not present to provide a proof of existence. Playfair's version of the axiom has become so popular that it is often referred to as Euclid's parallel axiom,[5] even though it was not Euclid's version of the axiom. A corollary of the axiom is that the binary relation of parallel lines is a serial relation.