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 eighteenth century produced more sophisticated proofs and although not correct, produced developments that were later used in non-Euclidean geometry. 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. 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. Proclus and Ptolemy also published some attempts to prove the parallel postulate. 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. Aristotle's treatment of the parallel postulate was lost. Several Greek scientists and mathematicians considered the parallel postulate after the appearance of Euclid's Elements, around 300 b.c.

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. For an estimated 22 centuries, Euclidean geometry held its weight.ĭespite 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 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. 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. Postulate II allows for lines of infinite length, which are denied in Elliptic geometry, where only finite lines are assumed.Įuclid was thought to have instructed in Alexandria after Alexander the Great established centers of learning in the city around 300 b.c. Euclid's parallel postulate may also be stated as one and only one parallel to a given line goes through a given point not on the line.Įlliptic geometry uses a modification of Postulate II. Hyperbolic geometry is based on changing Euclid's parallel postulate, which is also referred to as Euclid's fifth postulate, the last of the five postulates of Euclidian Geometry. Although there are different types of Non-Euclidean geometry which do not use all of the postulates or make alterations of one or more of the postulates of Euclidean geometry, hyperbolic and elliptic are usually most closely associated with the term non-Euclidean geometry. If a transversal falls on two lines in such a way that the interior angles on one side of the transversal are less than two right angles, then the lines meet on the side on which the angles are less than two right angles.Ī consistent logical system for which one of these postulates is modified in an essential way is non-Euclidean geometry.All right angles are equal to one another.A circle may be described with any point as center and any distance as a radius.A finite straight line can be produced continuously in a straight line.A straight line can be drawn from any point to any point.The first five postulates of Euclidean geometry will be listed in order to better understand the changes that are made to make it non-Euclidean. These geometries deal with more complex components of curves in space rather than the simple plane or solids used as the foundation for Euclid's geometry. There are other types of geometry which do not assume all of Euclid's postulates such as hyperbolic geometry, elliptic geometry, spherical geometry, descriptive geometry, differential geometry, geometric algebra, and multidimensional geometry. Non-Euclidean geometry refers to certain types of geometry which differ from plane and solid geometry which dominated the realm of mathematics for several centuries.

Non euclidean geometry art