In about 300 BC Euclid wrote The Elements, a book that became one of the most famous books ever written. Euclid stated five postulates which he based all of his theorem. Euclid's fifth postulate, which reads:"If two lines are drawn so that they intersect a third in such a way that the sum of interior angles on one side less than two right angles, then those two lines, if extended far enough, must intersect each other on a particular side".It is clear that the fifth postulate is different from the other four.That does not satisfy Euclid and he tried to avoid its use as long as possible, in fact the first 28 propositions of The Elements proved without using it. Another comment that appears at the moment is that of Euclid and many pengikutinya, assume that the straight line is infinite.
Proclus (410-485) wrote a comment on The Elements where he comments on the evidence to try to deduce the fifth postulate from the other four, in particular he notes that Ptolemy had produced evidence of 'false'. Proclus then goes on to give false evidence of his own. But he did not give the following theorem which himpunanara with the fifth postulate.Playfair axiom: "Given a line and a point not on that line, it is possible to draw exactly one line through a point parallel to the line."Although well known from the time of Proclus, this became known as the Axiom himpunanelah Playfair John Playfair wrote the famous commentary on Euclid in 1795 in which he proposed replacing Euclid's fifth postulate with these axioms.Many attempts were made to prove the fifth postulate from the other four, many of them are accepted as evidence for a period of time until the error was discovered. Always a mistake that's assuming some 'obvious' property that was himpunanara with the fifth postulate. evidence 'One such' given by Wallis in 1663 when he thought he had concluded the fifth proposition, but he really shows it is himpunanara with:"To himpunaniap triangle, there is a similar triangle of arbitrary magnitude."
One proof attempt was more important than most others. It was produced in 1697 by Girolamo Saccheri. The importance of Saccheri's job is that he is considered the fifth postulate false and attempted to get a contradiction.Here is a Saccheri quadrilateral
In figure Saccheri proved that the angle of the peak in the D and C is a testament to equal.The using the properties of congruent triangles which Euclid proved in Proposisi4 and 8 are proved before the fifth postulate digunakan.Saccheri have shown:a) The apex angle is> 90 ° (the hypothesis of an obtuse angle).b) apex angle is <90 ° (the hypothesis of acute angle).c) is the apex angle = 90 ° (the hypothesis of right angle).
Euclid's fifth postulate is c). Saccheri proved that the hypothesis of obtuse angle implied the fifth postulate, thus getting a contradiction. Saccheri then studied the acute angle hypothesis and many theorems are derived from non-Euclidean geometry without realizing what he was doing. But he finally 'prove' that the hypothesis of acute angle causes a contradiction with the assumption that there is a 'point at infinity' which is located in the area.
In 1766 Lambert followed a similar line to Saccheri. But he did not fall into the trap that Saccheri fell into and investigate hypotheses acute angle without obtaining a contradiction.Lambert noticed that, in this new geometry, the number of triangles increases as the angle decreases the triangle area.Legendre spent 40 years of his life working on the parallel postulate and the work appears in the annex to the various editions of its geometry very successful book Elements de Géométrie. Legendre proved that Euclid's fifth postulate is set by the number of angle triangle equals two right angles. Legendre showed, Saccheri has more than 100 years earlier, that the number of angle triangle can not be more than two right angles.This, once again like Saccheri, rested on the fact that an infinite straight line. In trying to show that the angle can not be less than 180 °, Legendre assumed that through a set of points in the interior angles is always possible to draw a line that meets both sides of the angle. This turned out to be another form himpunanara with the fifth postulate, but Legendre never realize his own mistakes.
Basic geometry is currently inundated with problems in the parallel postulate. D'Alembert, in 1767, called it scandalous basic geometry. The first person to really come to understand parallel problems is Gaussian. He began working on the fifth postulate in 1792 while only 15 years old, at first trying to prove the parallel postulate from the other four. In 1813 he had made little progress and writes:"In the theory of parallels we are even now not further than Euclid.This is a shameful part of mathematics ... "
But with 1817 Gauss had become convinced that the fifth postulate is independent of the other four postulates. He began working out the consequences of geometry in which more than one line can be drawn through a point parallel to a given line.Perhaps most surprising of all, Gauss never published this work, but keep it secret. At the time thought was dominated by Kant who has stated that Euclidean geometry is the inevitable requirement of thought and Gauss disliked controversy.
Gauss discussed the theory of parallels with his friend, the mathematician Farkas Bolyai who made false evidence several parallel postulate. János Bolyai Farkas Bolyai taught his son mathematics. In 1823 János Bolyai wrote to his father, saying that he knew that the Gaus has found the problem before but did not publish it. János Bolyai took two years to publish his book.
Work Bolyai reduced because Lobachevsky published a work on non-Euclidean geometry 1829. Neither Bolyai nor Gauss knew Lobachevsky work, mainly because only published in Russian in the Kazan Messenger a local university publications.Lobachevsky fared no better than Bolyai in gaining public recognition of the importance of work. He published an investigation on the geometric theory of parallel in 1840 that in 61 pages, provide the clearest record of the work of Lobachevsky.
Publishing accounts in France in Crelle's Journal in 1837 brought his work in non-Euclidean geometry mathematics community wide audience but not ready to accept ideas so revolutionary.In 1840 he Lobachevsky booklet explains clearly how the non-Euclidean geometry works."All of a straight line in the exit plane of the point can, by referring to the straight line given in the same plane, divided into two classes - into cutting and non-cut. This line of one and another class of the line will be called parallel to a given line. "
Here is a diagram of Lobachevsky
Hence Lobachevsky has replaced the fifth postulate of Euclid with Lobachevsky Parallel Postulate:"There are two parallel lines with a given line through a given point not on the phone."
Sources:
http://www-history.mcs.st-and.ac.uk/HistTopics/Non-Euclidean_geometry.htmlhttp://en.wikipedia.org/wiki/Non-Euclidean_geometry
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