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.

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.

5C's:+

ReplyDeleteWhat's your takeaway from this?

I think that it was great (and encouraging) of you to take on the task of explaining Euclid's parallel postulate. It is the first topic I try to stay away from! But, after reading your post I did understand it a lot more, though I'm still steering clear of it (just kidding)! I also like how you brought other mathematicians into it.

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