Axiom:Playfair's Axiom

Axiom

Exactly one straight line can be drawn through any point not on a given line parallel to the given straight line in a plane.

Or:

Given any straight line and a point not on it, there exists one and only one line which passes through this point and does not intersect the first line no matter how far they are extended.

This unique line is defined as being parallel to the original line in question.

Or:

Two straight lines which intersect one another cannot both be parallel to one and the same straight line.

Source of Name

This entry was named for John Playfair.

Historical Note

Playfair's axiom was not actually originated by John Playfair. He merely published it.

However, when he did so, he credited others, specifically William Ludlam, for having used it earlier.

It is a frequently seen alternative presentation of Euclid's Fifth Postulate.

It can easily seen to be equivalent to that given by Euclid, and it can be argued that it is easier to understand.

Texts on analytic geometry often gloss over its intrinsic significance, merely deducing it from the general equation of a straight line and the point of intersection of two such.

Sources

• 1926: Sir Thomas L. Heath: Euclid: The Thirteen Books of The Elements: Volume 1 (2nd ed.) ... (previous) ... (next): Book $\text{I}$. Notes on Postulate $5$
• 1933: D.M.Y. Sommerville: Analytical Conics (3rd ed.) ... (previous) ... (next): Chapter $\text {II}$. The Straight Line: $9$. Parallel lines. Points at infinity
• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): Euclid's axioms
• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): parallel postulate
• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): Playfair's axiom
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Playfair's axiom