Later mathematicians either tried to prove it or replace it by a simpler
postulate. In 1795, John Playfair stated it in a form which gained popularity: for
a line l and a point P not on the line l there is a unique line passing through P
parallel to l. Around the same time, Adrien Marie Legendre substituted another
equivalent version when he asserted the existence of a triangle whose angles add
up to 180 degrees. These new forms of the fifth postulate went some way to
meet the objection of artificiality. They were more acceptable than the
cumbersome version given by Euclid.
Another line of attack was to search for the elusive proof of the fifth postulate.
This exerted a powerful attraction on its adherents. If a proof could be found, the
postulate would become a theorem and it could retire from the firing line.
Unfortunately attempts to do this turned out to be excellent examples of circular
reasoning, arguments which assume the very thing they are trying to prove.
Non-Euclidean geometry
A breakthrough came through the work of Carl Friedrich Gauss, János Bolyai
and Nikolai Ivanovich Lobachevsky. Gauss did not publish his work, but it seems
clear he reached his conclusions in 1817. Bolyai published in 1831 and
Lobachevsky, independently, in 1829, causing a prority dispute between these
two. There is no doubting the brilliance of all these men. They effectively showed
that the fifth postulate was independent of the other four postulates. By adding
its negation to the other four postulates, they showed a consistent system was
possible.