For hundreds of years, they sought to prove that any alternative to the single parallel line model was impossible they really thought that if the parallel postulate was violated and more than one parallel was allowed, logical chaos would ensue.
Khayyam then replaces Euclid's 29th proposition, the first in which the parallel postulate is used, with a new sequence of eight propositions.
Khayyam's Explanation of the Difficulties in the Postulates of Euclid15 deals with the two most important issues in this context, the parallel postulate and the definition of equality of ratios.
Familiar congruence theorems follow in each edition until the parallel postulate could no longer be ignored.
The metrical account led to a progressive elucidation of a significant obscurity in Euclid's Elements: the parallel postulate.
Firstly, the discovery of non-Euclidean geometry in the early 19th Century showed that apparent self-evidence, at least in the case of the Parallel Postulate, is no guarantee of necessary truth.
Investigations of the parallel postulate began in Greek times, continued in the Islamic world, and were undertaken in the early modern West.
The construction of squares requires the immediately preceding theorems in Euclid, and depends upon the parallel postulate.
In a notable addition to Euclid, he tried valiantly to prove the parallel postulate (discussed later in Non-Euclidean geometries).
They differ largely, as he had to admit, in their unsuccessful attempts to deduce the parallel postulate.
