Around 300 BC, Euclid of Alexandria began a seemingly impossible task i.e. to summarise all of mathamatics known at the time. For about 15-20 years, Euclid devoted his time and energy to the discipline and composed “The elements”, the go to maths textbook for the next 2,000 years! In this book of his,Euclid established the use of the axiomatic method which is a method in which certain postulates are assumed(e.g. if u cut a line segment in the middle, it will split into two equal parts)to be correct and those postulates are used to develop the mathamatics. The book contained all the discoveries in number theory and geometry of the time. In the elements,Euclid assumed 5 postulates and 5 common notions. Amongst the five postulates the 5th postulate(also known as the parallel postulate)turned out to be quite intriguing, which is➖
If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if extended indefinitely, meet on that side on which the angles are less than two right angles
“OR”
Given a line and a point not on that line, there is exactly one line through the point that is parallel to the original line.
(Note:-The 2nd interpretation of the postulate is what's discussed in this paper as this interpretation is necessary for understanding non-euclidian geometry)
2nd postulate:- “A straight line segment can be extended indefinitely in both directions.
(Note:- This paper will only deal with the 5th and 2nd postulate as in order to understand non-euclidian geometry only those two are needed, for some basic understanding that is)
Now, at first the postulate may seem daunting because that is how the mathematicians felt. The first 4 postulates were very short and blatantly obvious but then you have a statement which is a paragraph long and for which there is no proof technically speaking
Various mathematicians like Ptolemy and Proclus thought that they had succeeded in proving the parallel postulate but they had only rephrased it in a number of ways.
{Images of Ptolemy(left) and Proclus(right)
{Proclus’s remark on the parallel postulate}
Here is an attempt made by Andrien Marie Legendre.( in the proof the postulate,assumptions were made by Legendre that the the first four postulates except the fifth postulate is true so that one can prove the 5th postulate. It also assumes that the 5th postulate is provable)
(Source:- Euclidean and non-Euclidean geometries: history and development by Marvin Jay Greenberg)
What mathematicians tried to do was they assumed that the parallel postulate was false and they replaced it with another postulate and tried to prove theorems with it if there was a contradiction then the parallel postulate is correct. This approach was taken by thinkers like al- Kharizimi and various others(though it is a very lack luster and impossible approach to the postulate)