1. The Parallel Postulate
Wei Cheng Liu
Cluster 6

2. Introduction
Euclid is a mathematician who contributed greatly to the field of geometry. In his book, Euclid’s Elements, he proposed many postulates as well as theorems that people still use today. The book is claimed to be the most successful mathematical textbook as it survived for more than 23 centuries.

3. Axioms
Between any two distinct points, there exists a unique straight line.
For any straight line segment, there exists a unique extension.
For every point, there exists a unique circle of fixed radius.
All right angles are equal.

4. 5th Axiom (The Parallel Postulate)
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 produced indefinitely, meet on the side on which are the angles less than two right angles.
One and only one straight line may be drawn through any point P in the plane parallel to a given straight line AB.
There exists a pair of similar triangles. For example, triangles which are not congruent, but have the three angles of one equal, respectively, to the three angles of the other.

5. What is the problem?
Using just 5 axioms, Euclid derived 467 theorems. These axioms acted as the building blocks for his work. The issue with the 5th axiom is that it isn’t self evident and it hasn’t been proven. It is said to be independent from the other axioms, meaning it cannot be proven using the other four axioms. If the 5th axiom cannot be proven, then there is a chance for it to be false, suggesting that all the theorems that includes the 5th axiom are incorrect.

6. Proof by Proclus
Although it seems perfectly logical, there is a problem with this proof because it takes the assumption that parallels are equally distant everywhere. The theorem that says parallel lines are always equally distant is derived from the parallel postulate, therefore this proof indirectly assumed the 5th axiom.
Let AB and CD be two parallel lines and a straight line EF cutting AB at E. Let a point P be on the line of EF. As P moves along EF towards F, the length perpendicular from P to AB becomes greater and eventually becomes greater than the distance between the parallel lines. Therefore, EF must intercept CD. A E B P P’ F C D

7. Proof by Nasiraddin A B C D
Nasiraddin began by making the assumption that if two straight lines, AB and CD, are related so that perpendicular lines drawn from CD to AB will always make unequal angles with AB, which the angles are always acute towards B and obtuse towards A. This suggests that the lines converge in the direction of B and D and diverge in the direction of A and C. A B C D

8. Proof by Nasiraddin(continued)
Then comes the proof that angles CDA and DCB are right angles by using the assumption Nasiraddin made. The reasoning is that if DCB is acute, then DA would be shorter than CB, but we know that DA and CB were equal distance so DCB is not acute. By the same logic DCB is not obtuse. Therefore, both DCB and CDA are right angles by the same logic.
Let AB be a segment and then draw perpendiculars of equal length AD and BC on the same side. Connect C to D. D C A B

9. Proof by Nasiraddin(continued)
Nasiraddin proved that there is only one way to draw a parallel line from a line and an external point. However, the flaw within this proof is the assumption made in the beginning, which is that two lines will converge if a perpendicular line drawn to one of the two lines will make unequal angles. This assumption is not acceptable without the proof of the 5th axiom.

10. Proof by Wallis E
Let lines AB and CD be cut by the transversal EF in point G and H and with the sum of angles BGH and DHG less than two right angles. It can be shown that angle EGB is larger than angle GHD. Then, if segment HG is moved along EF, with HD attached to it, point H will eventually reach G. HD takes the position of GI and is above GB. This means that HD must cut GB. A
The fallacy of this proof is that Wallis used the assumption that given a triangle, it is possible to make similar triangles. This assumption is equivalent to the fifth postulate, therefore rendering this proof invalid. G I B C H D F

11. More problems with proves
Some of the other fallacies include the use of non- Euclidean space such as the surface of a sphere or the hyperbolic plane. The reason why those planes are not accepted for prove is because the 5th postulate does not hold true in these planes. There is no way to draw 2 parallel lines in a sphere and there are infinite ways to draw 2 parallel lines in the hyperbolic plane. The 5th postulate stands to be very important to geometry but without any valid prove, we can only believe that the parallel postulate is an assumption.

12. Bibliography
Wolfe, Harold Eichholtz. Introduction to Non-Euclidean Geometry. New York: Dryden, Web.
Grabiner, Judith V. "Why Did Lagrange "Prove" the Parallel Postulate?" Am Math Mon American Mathematical Monthly (2009): Mathematical Association of America, Jan Web. 26 July 2015.
Mlodinow, Leonard. Euclid's Window: The Story of Geometry from Parallel Lines to Hyperspace. New York: Free, Print.
Clark, David M. Euclidean Geometry: A Guided Inquiry Approach. Berkeley, CA: Mathematical Sciences Research Institute, Print.

13. THANK YOU!!
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