# Extra Credit Paper

Due Tuesday, December 20, 2016

Write a paper (up to 5 pages double spaced) on the history of Euclid’s most controversial axiom, the so-called “Parallel Postulate”. This paper need not be a formal research paper (in the sense of following any particular format), but do cite your sources (internally and with a references page at the end). There is no requirement for having a number of sources.

First, you must explain what the fifth postulate actually says. Ideally, one should include a drawing which would show intuitively what it says. You should explain why it was considered controversial; that is, why did many mathematicians, throughout history, believe that this axiom is somehow different from Euclid’s other axioms? Why did they believe that this statement should not be considered as an axiom, but rather that we should try to prove it from some other axioms?

There were many attempts to resolve this controversy. Many mathematicians, over time, proposed other axioms which they felt would be simpler, but in fact ended up being logically equivalent to the Parallel Postulate. Write about some of these axioms and the mathematicians who came up with them. Make sure to explain why they believed these axioms were simpler and give some insight as to what the intuition for these axioms are as well (you may wish to draw a picture here as well).

Other mathematicians attempted to resolve the controversy by looking for a “proof by contradiction”. That is, they assumed that the Parallel Postulate was false, and they looked for a contradiction. This effort, while unsuccessful, ended up being very fruitful in showing that the Parallel Postulate is independent of the other axioms. That is, modern mathematicians accept that we can study geometries in which this axiom fails, and in fact this axiom can fail in a couple of ways. These geometries are called “elliptical” and “hyperbolic” (together, they are examples of “non-Euclidean geometries”). Briefly describe the history of these geometries: when were they first studied (in particular, do look for examples of unsuccessful proofs by contradiction of the Parallel Postulate), which mathematicians introduced them, and intuitively, what the difference between these geometries and Euclidean geometry is.

The amount of extra credit awarded is based on the following factors:

Background: (up to 1 points) Did you clearly state Euclid’s fifth postulate, along with giving some intuition on why one might accept it? Did you give an idea of why this is considered “controversial”?

Alternate Axioms: (up to 2 points) Did you find other axioms equivalent to the Parallel Postulate? Did you attribute the axioms to the mathematician who introduced it, and did you give an intuitive understanding of why this mathematician might have accepted this axiom?

Alternate Geometries: (up to 2 points): Did you give some intuitive explanation of elliptical and hyperbolic geometries? Did you describe the history of these geometries? Be careful here: these geometries were not created and accepted immediately. Sometimes, they were studied in one time period (perhaps the Middle Ages in Arabia), and then ultimately accepted in another time period (perhaps 1800s Europe). If that is the case, then describe both who discovered it and how it was rediscovered later on.

The amount of extra credit you earn (up to 5 points) will be added to the grade you receive on your final exam.