Axiom:Euclid's Fifth Postulate/Historical Note

Historical Note on Euclid's Fifth Postulate
During the course of the history of mathematics, many mathematicians have attempted to prove Euclid's fifth postulate from the remaining four, as it seems less intuitive than the rest of Euclid's Postulates.

himself avoided using the fifth postulate until the $29$th proposition (part $1$): Parallelism implies Equal Alternate Interior Angles in his seminal.

Eventually, in $1823$ and, independently in $1832$,, realized that self-consistent, non-euclidean geometries could be developed by not accepting this postulate.

It transpired that himself had already come to a similar conclusion, but had not had the confidence to publish.

In fact, in a letter to, after receiving a copy of ' appendix, wrote:
 * If I commenced by saying that I am unable to praise this work, you would certainly be surprised for a moment. But I cannot say otherwise. To praise it, would be to praise myself. Indeed the whole contents of the work, the path taken by your son, the results to which he is led, coincide almost entirely with my meditations, which have occupied my mind partly for the last thirty or thirty-five years. So I remained quite stupefied. So far as my own work is concerned, of which up till now I have put little on paper, my intention was not to let it be published during my lifetime. ... It is therefore a pleasant surprise for me that I am spared this trouble, and I am very glad that it is just the son of my old friend, who takes the precedence of me in such a remarkable manner.

Later, in a letter to in $1829$,  wrote:
 * It may take very long before I make public my investigations on this issue: in fact, this may not happen in my lifetime for I fear the "clamor of the Boeotians."

Hence it is apparent that the fifth postulate is in fact axiomatic for Euclidean geometry.

The literature on this subject is voluminous, but since the birth of the concept of non-Euclidean geometry and (to a certain extent) since the study of analytic geometry developed, most of this literature has little mathematical value now beyond historical interest.