Aristotle's axiom explained
Aristotle's axiom is an axiom in the foundations of geometry, proposed by Aristotle in On the Heavens that states:
If
is an acute angle and AB is any segment, then there exists a point P on the ray
and a point Q on the ray
, such that PQ is perpendicular to OX and PQ > AB.
Aristotle's axiom is a consequence of the Archimedean property, 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.
Without the parallel postulate, Aristotle's axiom is equivalent to each of the following three incidence-geometric statements:
- Given a line A and a point P on A, as well as two intersecting lines M and N, both parallel to A 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.
- Given a line A and two distinct intersecting lines M and N, each different from A, there exists a line G which intersects A and M, but not N.
