Bolyai and Lobachevsky constructed a new geometry by allowing there to be
more than one line through P that does not meet the line l. How can this be?
Surely the dotted lines meet l. If we accept this we are unconsciously falling in
with Euclid’s view. The diagram is therefore a confidence trick, for what Bolyai
and Lobachevsky were proposing was a new sort of geometry which does not
conform to the commonsense one of Euclid. In fact, their non-Euclidean
geometry can be thought of as the geometry on the curved surface of what is
known as a pseudosphere.
The shortest paths between the points on a pseudosphere play the same role
as straight lines in Euclid’s geometry. One of the curiosities of this non-Euclidean
geometry is that the sum of the angles in a triangle is less than 180 degrees. This
geometry is called hyperbolic geometry.
Another alternative to the fifth postulate states that every line through P meets
the line l. Put a different way, there are no lines through P which are ‘parallel’ to
l. This geometry is different from Bolyai’s and Lobachevsky’s, but it is a genuine