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 first four axioms that at least one parallel line exists given a line L and a point P not on L, as follows:
- Construct a perpendicular: Using the axioms and previously established theorems, you can construct a line perpendicular to line L that passes through P.
- Construct another perpendicular: A second perpendicular line is drawn to the first one, starting from point P.
- Parallel Line: This second perpendicular line will be parallel to L by the definition of parallel lines (i.e the alternate interior angles are congruent as per the 4th axiom).
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 one" 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.
- ^ Playfair 1846, p. 29
- ^ more precisely, in the context of absolute geometry.
- ^ Euclid's elements, Book I, definition 23
- ^ Heath 1956, Vol. 1, p. 190
- ^ for instance, Rafael Artzy (1965) Linear Geometry, page 202, Addison-Wesley