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.

Playfair's axiom is a geometrical axiom, intended to replace the fifth postulate of Euclides (the Parallel postulate):

Given a line and a point not on it, at most one parallel to the given line can be drawn through the point.

It is equivalent to Euclid's parallel postulate and was named after the Scottish mathematician John Playfair. It is only required to state "at most" because the rest of the postulates will imply that there is exactly one. It could perfectly be assumed to write it saying "there is one and only one parallel". It is important to remark that in the Euclid book, two lines are said to be parallel if they never meet. It does not matter if their distance is always the same or not.

When David Hilbert made his Hilbert's axioms he used Playfair's axiom instead of the original one from Euclid.

This axiom is used not only in Euclidean geometry, but also in a broader study called affine geometry where the concept of parallelism is central. In the context of affine geometry the axiom has been called Euclid's parallel axiom, but for Euclidean geometry the parallel postulate which refers to angles is the traditional expression of parallelism.

History

In 1795 John Playfair published an alternative, more stringent formulation of Euclid's parallel postulate, which is now called Playfair's axiom; though the axiom bears Playfair's name, he did not create it, but credited others, in particular William Ludlam, with the prior use of it. However, Proclus (410–485 A.D.) clearly makes the statement in his commentary on Euclid I.31 (Book I, Proposition 31)

Relation with Euclid's fifth postulate

If the sum of the interior angles α and β is less than 180°, the two straight lines, produced indefinitely, meet on that side.

Playfair's axiom is not exactly equivalent to Euclid's Fifth Postulate because on an elliptical geometry, such as the surface of a sphere, Euclid's postulate in its original version:

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.

holds because two lines on a sphere always meet. Playfair's postulate is therefore stronger and prevents elliptic geometries. Therefore, it is not possible to derive Playfair's postulate from Euclid Fifth alone.

Euclid's fifth postulate with the other four implies Playfair's postulate

The easiest way to show this is using the Euclid theorem (based in the fifth postulate) that states that the angles of a triangle are two right angles. Given a line and a point, construct a line perpendicular to the given one by the point, and a perpendicular to the perpendicular. This line is parallel because it cannot form a triangle. Now it can be seen that no more parallels exist because any line that forms an angle with the second one will cut the first one.

Playfair's 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 are less than two right angles, but this is more difficult.

References

• Heath, Thomas L. (1956). The Thirteen Books of Euclid's Elements (3 vols.) (2nd ed. [Facsimile. Original publication: Cambridge University Press, 1925] ed.). New York: Dover Publications. ISBN 0-486-60088-2 (vol. 1), ISBN 0-486-60089-0 (vol. 2), ISBN 0-486-60090-4 (vol. 3) .  Heath's authoritative translation plus extensive historical research and detailed commentary throughout the text.