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 oth

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.

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: Euclide

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Approximate Word count = 680
Approximate Pages = 3 (250 words per page double spaced)