Aristotle's axiom

Aristotle's axiom is an axiom in the foundations of geometry, proposed by Aristotle in On the Heavens that states:

Aristotle's axiom asserts that a line PQ exists which is parallel to AB but greater in length. Note that: 1) the line AB does not need to intersect OY or OX; 2) P and Q do not need to lie on the lines OY and OX, but their rays (i.e. the infinite continuation of these lines).

If ${\displaystyle {\widehat {\rm {XOY}}}}$ is an acute angle and AB is any segment, then there exists a point P on the ray ${\displaystyle {\overrightarrow {OY}}}$ and a point Q on the ray ${\displaystyle {\overrightarrow {OX}}}$, such that PQ is perpendicular to OX and PQ > AB.

Aristotle's axiom is a consequence of the Archimedean property,^[1] and the conjunction of Aristotle's axiom and the Lotschnittaxiom, which states that "Perpendiculars raised on each side of a right angle intersect", is equivalent to the Parallel Postulate.^[2]

Without the parallel postulate, Aristotle's axiom is equivalent to each of the following two incidence-geometric statements:^{[3] [4]}

• Given two intersecting lines m and n, and a point P, incident with neither m nor n, there exists a line g through P which intersects m but not n.
• Given a line a as well as two intersecting lines m and n, both parallel to a, there exists a line g which intersects a and m, but not n.