’s 5Th Postulate Essay, Research Paper
Euclid s Fifth Postulate
One of the most fascinating aspects of Mathematics is that there exist statements that are both true and false. Perhaps the most famous of these is Euclid s controversial fifth postulate. Throughout history, almost from the postulate conception, mathematics have tried to prove or disprove it. Actually, it seems that even Euclid himself did not entirely trust the postulate, for he avoided using it as long as he could in his great work, The Elements.
From the beginning, Euclid s fifth postulate, also called the parallel postulate stood out from among Euclid s other postulates. The first four postulates are short, brief, and to the point, whereas the fifth is longer and rather strange sounding. The postulates are listed in The Elements as such:
1. To draw a straight line from any point to another.
2. To produce a finite straight line continuously in a straight line.
3. To describe a circle with any center and distance.
4. That all right angles are equal to each other.
5. That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, if produced infinitely, meet on that side on which are the angles less than the two right angles.
Clearly there are dissimilarities between the last and the first four, as Euclid must have been aware, mainly in the length of the postulate. In any case, mathematicians of the time, seeing that there was something strange about this parallel postulate set out to prove it by using only the first four postulates.
Many mathematicians, among them Poisonous, tried to define parallel lines in another way than Euclid. However, it was quickly observed that these definitions contained the fifth postulate as a premise. Needless to say, many great Greek mathematicians tried their hands at proving the postulate in a variety of ways; they all failed.
Later, the Arabic mathematicians, who succeeded the
Faulty attempts to prove the parallel postulate, though, were by no means only a problem of antiquity. One attempted proof by Sacceri in 1697 was particularly significant in that he assumed the postulate false and tried to reach a contradiction. In doing so, he unknowingly derived many of the theorems of non-Euclidean geometry.
The first to truly realize that there may be another form of geometry other than Euclidean, was Gauss, who began to study a geometry in which more than one line can pass through the same point and still be parallel to another in the early 19th century. Disliking controversy, he kept his studies to himself. In fact, he first to publicly claim that another geometry was possible was Janos Bolyai. He published his theory in an appendix to his father s book on geometry.
At the same time, in Russia, Lobaschevsky published a work on this alternate geometry in 1829. His work did not receive wide recognition, though, and he developed his theory a great deal more before it did so. In the Bolyai-Lobaschevian model, there are two lines which pass through a point not on a line parallel to that line. Similarly, Riemann, who studied under Gauss, mentioned a theory of spherical geometry in a lecture which was not published until after his death. In this geometry, no parallels are possible. It was at this time that mathematicians realized that the mystery of parallel lines would never be solved satisfactorily. Today, there are two main classes of geometry: Euclidean and Non-Euclidean, which consists of both the Bolyai-Lobaschevsky and Reimann models. It has been sufficiently proven that Euclid s fifth postulate can be both true and false. By assuming it true, one can generate the geometrical world know as Euclidean Geometry in which no contradiction has arisen. Likewise, by assuming a different theory of parallels, one can generate another, quite separate geometrical world in which no contradiction has been found.