The Parallel Postulate

Abstract

EUCLID'S famous postulate was responsible for an enormous amount of mathematical activity over period of more than twenty centuries. The failure of mathematicians to prove Euclid's statement from his other postulates con tributed to Euclid's fame and eventually led to the invention of non-Euclidean geometries. Before Euclid's time, various definitions of lines had been considered by the Greeks and then discarded. Among them were parallel lines are lines every where equidistant from one another and parallel lines are lines having the same direction from given But these early definitions were sometimes vague or contradictory. Euclid tried to overcome these difficulties by his definition, Parallel lines are straight lines which, being in the same plane and being produced indefinitely in both directions, do not meet one an other in either direction, and by his fifth postulate, Let it be postulated that, if 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 that side on wThich the angles are less than two right angles. The statements of Euclid's first four assumptions are: Let the following be postulated: (1) To draw straight line from any point to any point. (2) To pro duce finite straight line continuously in straight line. (3) To describe circle with any center and distance. (4) That all right angles are equal to one another. All of his assumptions fall into one of two categories. The first is the set of self evident facts concerning plane figures. An example of such an assumption is that a straight line is the shortest distance between two points. The second cate gory deals with concepts beyond the realm of actual experience. For example, Euclid stated that a straight line must continue undeviatingly in either direction without end and without finite length. Since it is impossible to experience things indefinitely far off, anything that is said about events there is speculation, not self-evident truth. The fifth postulate falls into this latter category. The complicated nature of the fifth postulate led numerous mathematicians to believe that it could be proved using the remaining postulates, and, therefore, ought to be theorem rather than pos tulate. Even Euclid might have supported this viewpoint since he did succeed in proving the converse of the postulate. One of the first geometers who attempted to prove statement equivalent to Euclid's postulate was Posidonius, in the first century b.c. He had defined lines as lines that are coplanar and equi distant. A second early important attempt to prove the postulate was made by Claudius Ptolemy of Alexandria, in the second century. In the fifth century, Proclus, who had studied mathematics in Alexandria and taught in Athens, worked extensively on the problem of proving Euclid's fifth postulate. He succeeded in showing that