In the previous page, we discussed some motivation for wanting to replace (1.1) with the more useful (1.2):

 

(1.1) Two distinct lines meet in at most one point.

(1.2) Two distinct lines meet in one and only one point.

 

In Euclidean Geometry, the main problem with making the stronger statement (1.2) is the case when two lines don’t meet at any point.  That is, when two lines are parallel.  One strategy for extending (1.1) to (1.2) would be to allow parallel lines to meet at exactly one point: say at infinity.

 

Parallel Lines intersecting?!?!?

 

Euclid’s 5th Postulate states that lines will always intersect at some point unless they are parallel.  However, this is an axiom, not a theorem.  In other words, Euclid just assumed this to be a geometric truth, without proof.  Many subsequent mathematicians believed this Postulate was independent of the other 4 Postulates; one could prove it as a Theorem using only the other Postulates.  However, nobody was ever able to complete such a proof, and in 1868, the mathematician Beltrami formally proved that the ‘Axiom of Parallels’ was completely independent of the other Postulates.

 

What does this mean?  Apart from Euclid’s Postulate, there is no guarantee that parallel lines cannot meet.  Thus the several varieties of ‘non-Euclidean’ Geometry (where parallel lines can meet) can be entirely consistent. 

 

Even before Beltrami proved the independence of the Parallel Postulate, mathematicians were still able to work on Projective Geometry.  In the early 17th Century, Kepler suggested the notion of ‘points at infinity’ where parallel lines would intersect; meanwhile Desargues and Pascal began to study Geometry using only intersections.  Once Kepler’s idea was taken seriously, Geometers saw that the Geometry of intersections (incidence relations) could be made into a wholly consistent theory.  As suggested above, if all lines are guaranteed to meet at one point, the study of intersections does not have to make any exceptions (a flaw of Euclidean Geometry).  Finally, in 1871, Klein proved that the entire theory of Projective Geometry is independent of the Parallel Postulate.

 

 

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