Non-Euclidean Geometry

Euclid was thought to have instructed in Alexandria after Alexander the Great established centers of learning in the city around 300 B.C. Euclid was the mathematician who collected all of the definitions, postulates, and theorems that were available at that time, along with some of his insights and developments, and placed them in a logical order and completed what we now know as Euclid's Elements.

The influence of Greek geometry on the mathematics communities of the world was profound for in Greek geometry was contained the ideals of deductive thinking with its definitions, corollaries, and theorems which could establish beyond any reasonable doubt the truth or falseness of propositions. For an estimated 22 centuries, Euclidean geometry held its weight.

Despite the general acceptance of Euclidean geometry, there appeared to be a problem with the parallel postulate as to whether or not it really was a postulate or that it could be deduced from other definitions, propositions, or axioms. The history of these attempts to prove the parallel postulate lasted for nearly 20 centuries, and after numerous failures, gave rise to the establishment of Non-Euclidean geometry and the independence of the parallel postulate.

Several Greek scientists and mathematicians considered the parallel postulate after the appearance of Euclid's Elements, around 300 B.C. Aristotle's treatment of the parallel postulate was lost. However, it was the Arab scholars who appeared to have obtained some information on the last text and reported that Aristotle's treatment was different from that of Euclid since his definition depended on the distance between parallel lines. Proclus and Ptolemy also published some attempts to prove the parallel postulate.

Omar Khayyam provided extensive coverage on the proof of the parallel postulate or theory of parallels in his discussions on the difficulties of making valid proofs from Euclid's definitions and theorems. During the thirteenth century Husam al-Din al-Salar wrote a text on the parallel postulate in an attempt to improve on the development by Omar Khayyam.

The eighteenth century produced more sophisticated proofs and although not correct, produced developments that were later used in non-Euclidean geometry. The Italian mathematician, Girolamo Saccheri, in one of his proofs considered non-Euclidian concepts by making use of the acute-angle hypothesis on the intersection of two straight lines.

The attempt to solve this problem was made also by Farkas Bolyai, the father of Johann Bolyai, one of the founders of non-Euclidean geometry but his proof was also invalid. It is interesting to note that Johann's father cautioned his son not to get involved with the proof of the parallel postulate because of its complexity.