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.
It is equivalent to Euclid's parallel postulate in the context of Euclidean geometry and was named after the Scottish mathematicianJohn 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. 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 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, 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
clearly makes the statement in his commentary on Euclid I.31 In 1785 William Ludlam expressed the parallel axiom as follows: This brief expression of Euclidean parallelism was adopted by Playfair in his textbook Elements of Geometry that was republished often. He wrote Playfair acknowledged Ludlam and others for simplifying the Euclidean assertion. In later developments the point of intersection of the two lines came first, and the denial of two parallels became expressed as a unique parallel through the given point. In 1883 Arthur Cayley was president of the British Association and expressed this opinion in his address to the Association: When David Hilbert wrote his book, Foundations of Geometry, providing a new set of axioms for Euclidean geometry, he used Playfair's form of the axiom instead of the original Euclidean version for discussing parallel lines.
If a line segment intersects two straight lines forming two interior angles on the same side that sum to less than two right angles, then the two lines, if extended indefinitely, meet on that side on which the angles sum to less than two right angles.
The complexity of this statement when compared to Playfair's formulation is certainly a leading contribution to the popularity of quoting Playfair's axiom in discussions of the parallel postulate. Within the context of absolute geometry the two statements are equivalent, meaning that each can be proved by assuming the other in the presence of the remaining axioms of the geometry. This is not to say that the statements are logically equivalent, since, for example, when interpreted in the spherical model of elliptical geometry one statement is true and the other isn't. Logically equivalent statements have the same truth value in all models in which they have interpretations. The proofs below assume that all the axioms of absolute geometry are valid.
The easiest way to show this is using the Euclidean theorem that states that the angles of a triangle sum to two right angles. Given a line and a point P not on that line, construct a line, t, perpendicular to the given one through the point P, and then a perpendicular to this perpendicular at the point P. This line is parallel because it cannot meet and form a triangle, which is stated in Book 1 Proposition 27 in Euclid's Elements. Now it can be seen that no other parallels exist. If n was a second line through P, then n makes an acute angle with t and the hypothesis of the fifth postulate holds, and so, n meets.
Playfair's axiom implies Euclid's fifth postulate
Given that Playfair's postulate implies that only the perpendicular to the perpendicular is a parallel, the lines of the Euclid construction will have to cut each other in a point. It is also necessary to prove that they will do it in the side where the angles sum to less than two right angles, but this is more difficult.
Transitivity of parallelism
Proposition 30 of Euclid reads, "Two lines, each parallel to a third line, are parallel to each other." It was noted by Augustus De Morgan that this proposition is logically equivalent to Playfair’s axiom. This notice was recounted by T. L. Heath in 1908. De Morgan’s argument runs as follows: Let X be the set of pairs of distinct lines which meet and Y the set of distinct pairs of lines each of which is parallel to a single common line. If z represents a pair of distinct lines, then the statement, is Playfair's axiom and its logically equivalent contrapositive, is Euclid I.30, the transitivity of parallelism. More recently the implication has been phrased differently in terms of the binary relation expressed by parallel lines: In affine geometry the relation is taken to be an equivalence relation, which means that a line is considered to be parallel to itself. Andy Liu wrote, "Let P be a point not on line 2. Suppose both line 1 and line 3 pass through P and are parallel to line 2. By transitivity, they are parallel to each other, and hence cannot have exactly P in common. It follows that they are the same line, which is Playfair's axiom."
