A paper which discusses the philosophical and logical problems contained in Euclid's 'Fifth Postulate' on planar geometry.
Written in 2002; 1,622 words; 3 sources; APA; $ 52.95
Paper Summary:
Euclid gave the world much of the information it has on planar geometry in his five postulates. The paper shows that while the first four are relatively easy to understand, the fifth one is very difficult in relation to the others. It is this fifth postulate that many people feel can never be proven. The paper discusses how there are those that say it is simply incorrect, those that say it's both true and false and others that say there is no possible way to prove it, and Euclid himself may have realized that the task was impossible. The author of the paper surmizes that if someday the fifth postulate is proven to be either true or false, and the decision is agreed upon, then it could change the way mathematics are done and the way geometry is looked at.
From the Paper:
"Theoretically it would be possible for the lines to move toward one another so slowly, because of the low degree of angle, that they take a huge amount of space to come together at the end. But is it possible to have such a slight angle that the lines are almost parallel? They would be so close to parallel at that point that the impression that they are drawing closer together wouldn't be noticed unless they were looked at over miles at one time. That must be possible, but they still must meet somewhere in infinity.
Perhaps Euclid was right and the lines do meet somewhere, but the angles can be so minute that the lines go on almost to infinity, and we don't have the capabilities to calculate just how far that is yet. Perhaps Euclid is wrong and lines will go on into infinity still never touching, but only being a hair's width apart. Mathematicians may never know, since they haven't discovered any way to prove Euclid's fifth postulate by now."
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