What does the most controversial postulate of Euclid means
What does the most controversial postulate of Euclid means?What is the origin of the non-euclidean Geometry? Why did some Mathematicians were against with the fifth postulate?Who are involved in the discovery of the non-euclidean geometry?How did the Mathematicians discover non-euclidean geometry? How did the non-euclidean geometry develop over time?
Considering Euclid’s Postulates, the first four postulates, or axioms, were simply stated, but the Fifth Postulate was quite different from the others. Postulate V (Which is also called the Parallel Postulate) states that “That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles. (Katz, chapter 1)”. Unfortunately, several mathematicians found flaws against his fifth postulate, and have been trying to find proofs. These flaws and lack of proofs on Euclid’s fifth postulate lead the mathematicians to discover the Non-Euclidian Geometry.
Literally, non-Euclidean geometry means different kind of geometry than Euclidean Geometry. As background for the appearance of this geometry, there were many polemics around the fifth postulate in mathematical history. From ancient Greek to the beginning of the 19th century, numerous mathematicians attempted to prove the fifth postulate as a theorem using the other four postulates. Dissimilar to others, the fifth postulate seems complex, complicated and non-selfevident as a basic assumption constructing the Euclidean geometry, and looks more like a theorem rather than a postulate. For this reason, the last postulate was questioned for over two thousand years. The formulation of this postulate looks too complex to be easily accepted as a postulate. Initially, mathematicians had thought that due to the completeness of the logical structure of Euclidean geometry, it was the only geometry in the world. Their approach to the parallel postulate was limited and narrow. This was the main reason why Non-euclidean Geometry was developed.
At the turn of the century, during those critical years in the evolution of geometry, the dominant figure in the mathematical world was Carl Friedrich Gauss (1777–1855). Naturally he took no small part in the development of the ideas which led to the discovery of the new systems of geometry. Few of the results of his many years of meditation and research on the problems associated with the Fifth Postulate were published or made public during his lifetime. It was also he who first called the new geometry Non-Euclidean. Gauss, traveling in the footsteps of Saccheri and Lambert, with whose books he may have been familiar, was still attempting to prove the Fifth Postulate by the reduction and absurdum method, but that he fully recognized the profound character of the obstacles encountered. It was during the second decade that he began the formulation of the idea of a new geometry, to develop the elementary theorems and to dispel his doubts.not only that but he also think of other ways to contradict Euclidian Geometry. one of these things is when Gauss thought about a curved two-dimensional surface, like the surface of a globe mapping our planet’s oceans and continents. The vertical lines of longitude all make angles of 90 degrees with the equator on the globe’s surface, yet by the time they reach the poles they have met. So the parallel postulate is incorrect on curved surfaces. Gauss realized that self-consistent non-Euclidean geometries could be constructed. He saw that the parallel postulate can never be proven, because the existence of non-Euclidean geometry shows this postulate is independent of Euclid’s other four postulates. Consequently, among his papers there is to be found a brief account of the elementary theory of parallels for the new geometry. We have already noted that one of the simplest substitutes for the Fifth Postulate is the so-called Playfair Axiom. In rejecting the Postulate Gauss, like Bolyai and Lobachewsky, chose to assume that through a point more than one parallel (in the sense of Euclid) can be drawn to a given line.
Bolyai was numbered by Gauss as one of his friends and is certain that during their time , the two continues to be at work at proving the fifth postulate until such time that the two cut the communication because Gauss focuses on some other matter. But this doesn’t mean that the battle against the fifth postulate has come to an end with Bolyai. In his attempts to prove the Fifth Postulate by denying it, Bolyai chose to regard that assumption in the form which we have already designated as Playfair’s Axiom, and which asserts that one and only one parallel line can be drawn through a given point to a given line. The denial of the Postulate then implies either that no parallel to the line can be drawn through the point or that more than one such parallel can be drawn. But, as a consequence of Euclid I, 27 and 28, provided the straight line is regarded as infinite, the former of the two implications must be discarded. Furthermore, if there are at least two parallels to the line through the point, then there must be an infinite number of parallels in the sense of Euclid. Johann Bolyai published nothing more, though he continued his investigations. Notes found among his papers show that he was interested in the further extension of his ideas into space of three dimensions and also in the comparison of his Non-Euclidean Geometry with Spherical Trigonometry. It was this latter comparison which led him to the conviction that the Fifth Postulate could not be proved.14 He was never thoroughly convinced, however, that investigations into space of three dimensions might not lead to the discovery of inconsistencies in the new geometry.
In 1823, Lobachevsky had completed the manuscript for a textbook on elementary geometry, a text which was never published. It is where he made the significant statement that no rigorous proof of the Parallel Postulate had ever been discovered and that those proofs which had been suggested were merely explanations and were not mathematical proofs in the true sense. Evidently he had begun to realize that the difficulties encountered in the attempts to prove the Postulate arose through causes quite different from those to which they had previously always been ascribed. The next three years saw the evolution of his new theory of parallels. It is known that in 1826 he read a paper before the physics and mathematics section of the University of Kasan and on that occasion suggested a new geometry in which more than one straight line can be drawn through a point parallel to a given line and the sum of the angles of a triangle is less than two right angles. Unfortunately the lecture was never printed and the manuscript has not been found. But in 1829–30 he published a memoir on the principles of geometry in the Kasan Bulletin, referring to the lecture mentioned above, and explaining in full his doctrine of parallels. This memoir, the first account of Non-Euclidean Geometry to appear in print, attracted little attention in his own country, and, because it was printed in Russian, practically none at all outside. Confident of the merit of his discoveries, Lobachewsky wrote a number of papers, more or less extensive, on the new theory of parallels, hoping thus to bring it to the attention of mathematicians all over the world.
The idea of such a new geometry shook mathematics and science to its foundations, and these three mathematicians pave the way for the future discoveries or development on Non-Euclidean Geometry.
Undeniably, there’s no permanent thing in this world, only change. Everything undergo development and progress. This development will be now a major explosion which will be adapted by the community even if some are against. In the field of Mathematics, development is inevitable. The very proof of this was the development of the non-Euclidean Geometry which brought up the new universe. Surprisingly, the discovery of Non-Euclidean Geometry was not made by one man, but independently by several Mathematicians in the different parts of the world. This hadhappened more than once in the history of mathematics and it will doubtless occur again. And at about the identical time the discovery of a logically unswerving geometry, in which the Fifth Postulate was denied, was made by Gauss in Germany, Bolyai in Hungary and Lobachevsky in Russia. It is good to know that distance and space are never and will never be a hindrance in achieving development and progress in life.
Non- Euclidean Geometry is indeed another field of Mathematics. It has its own demarcation, following propositions and rules which can only be applied in its own world. Amazingly, the different branches of the non-euclidean geometry have only a single origin which is the fifth postulate of Euclid, ‘If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough’.It will always be credited to Euclid for his work in geometry which paved a way for the non-euclidean geometry to arise.