Playfair's axiom

[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.

Proclus (410–485 A.D.) clearly makes the statement in his commentary on Euclid I.31 (Book I, Proposition 31).

He wrote[8] Playfair acknowledged Ludlam and others for simplifying the Euclidean assertion.

[9] In 1883 Arthur Cayley was president of the British Association and expressed this opinion in his address to the Association:[10] When David Hilbert wrote his book, Foundations of Geometry (1899),[11] 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.

and form a triangle, which is stated in Book 1 Proposition 27 in Euclid's Elements.

If n was a second line through P, then n makes an acute angle with t (since it is not the perpendicular) and the hypothesis of the fifth postulate holds, and so, n meets

[17] The classical equivalence between Playfair's axiom and Euclid's fifth postulate collapses in the absence of triangle congruence.

It was noted[19] by Augustus De Morgan that this proposition is logically equivalent to Playfair’s axiom.

If z represents a pair of distinct lines, then the statement, is Playfair's axiom (in De Morgan's terms, No X is Y) and its logically equivalent contrapositive, is Euclid I.30, the transitivity of parallelism (No Y is X).