Weekly 1: History of Math

Euclid’s Fifth Postulate is one of the most historically discussed postulates in the history of mathematics.  Many of the most famous and recognized mathematicians have worked on the success of this postulate.  Some failed, some succeeded, and some work that aided in its discovery was not recognized until after the mathematician’s life ended.  The amount of effort and intelligence that went into this postulate is something to be recognized.

           At first, Euclid created five postulates.  The first four postulates describe the properties in exactly one single line.  It is not until the fifth postulate that parallel lines are introduced.  Mathematicians thought that the parallel (fifth) postulate could use all four previous postulates to help prove it.  The reason for so much discussion over the fifth postulate was because unlike the first four, there were many more discrepancies in the fifth. Mathematicians worked on the postulate until they found something to fall through in their work.

            The infamous Fifth Postulate is also known as the Parallel Postulate because it can draw conclusions about the characteristics of parallel lines.  This postulate was necessary to prove Proposition 29.  Euclid’s first 28 propositions could be proven with his first four.  Therefore, no further propositions, including parallel lines, could be proved without the understanding and correctness on the Fifth Postulate.  Interestingly, in Euclid’s original writings of the postulate, he assumes the creation of the parallel lines.  This did not sit well with other mathematicians, so the work began.

            There were many consequences that were created by the fifth postulate.  Many of these consequences looked to help conquer the ambiguity contained in the postulate.  They looked to observe the possibilities and behavior of the interaction between two lines.  It was not until the work of Saccheri and his approach to finding the answer.  Instead of trying to prove what Euclid had written, he wanted to prove that there existed a contradiction.  Although his work did not arrive at a concrete conclusion of contradiction, his method of using quadrilaterals was not completely a waste.  

            Following Saccheri were mathematicians looking for answers, but never found any.  The next big attempt came from Bernhard Riemann.  In Euclid’s Fifth Postulate, the length of the line is not defined and many people assumed that it meant infinite length.  Riemann found that an undefined length did not necessarily mean that the length was infinite.  This led to the discovery Elliptic Geometry.  Using a three dimensional plane, postulates 1, 2 and 5 were modified using circles.  In Elliptic Geometry, all the perpendiculars have equal length and the angles’ sum is determinable unlike plane geometry.  Riemann’s discoveries used a new and unique approach to geometry. 

            All of the failures and successes of mathematicians led to many discoveries.  Some work may have made accidental discoveries on the way, but without unique points of view, geometry would not be dynamic.  It took the work of Euclid, Riemann, Lobachevsky, Bolyai, and Gauss to challenge the work of others and make lasting discoveries.  Euclid laid most of the groundwork when it came to these discoveries.  If it was not for his work, he would have not paved the way for the other mathematicians.    Euclid’s Fifth Postulate created frustration for years, but without this frustration and determination, geometry would not have been redefined. 

The main message that I can take away from Euclid's work is that persistence is key.  He may not have been able to finish proving his work like he wanted to, but he paved the way for other mathematicians.  We need to understand that even mathematical tasks are done better by teams.  Math is not meant for one genius to try and conquer.  If you look at the other historical details of mathematics, you will see that multiple people have done their part to contribute to certain discoveries.  Yes, we know Euclid's name because of the work, successes, and troubles he had.  Experiencing difficulties only proves that he is human, which might have been a question I have had before.  

"Euclid as the father of geometry." Khan Academy. N.p., n.d. Web. 12 May 2014. <http://www.khanacademy.org/math/geometry/intro_euclid/v/euclid-as-the-father-of-geometry>.

"Parallel Postulate." - AoPSWiki. N.p., n.d. Web. 12 May 2014. http://www.artofproblemsolving.com/Wiki/index.php/Parallel_Postulate