Differences Between Neutral Geometry and Euclidean Geometry

Around 300 B.C. the first axioms of geometry were written by Euclid. He titled his works the Elements. The last of his theorems is the parallel postulate which states that for every line and for every external point, there exists a unique line through the external point that is parallel to the given line.

Neutral geometry has no dependence on the parallel postulate. This type of geometry consists of theorems that use only Euclid’s first four postulates and the axioms that can be derived from those postulates for proving their existence. While trying to prove Euclid’s 5th postulate, many mathematicians tried to negate postulate 5 to try to find discrepancies or inconsistencies. This allowed them to take a different look at the universe which resulted in the creation of a new geometry to describe space where are angles are equal and more than one parallel line can be drawn through a point that is not on the line. This geometry was called hyperbolic geometry.

For spherical geometry, which was developed so one could study figures on the surface of a sphere, the 5th postulate was not needed. This geometry primarily uses the second and fourth postulates. Euclid’s second postulate which states lines can be extended indefinitely in two directions can be altered and substitute the fifth postulate with a postulate called the Ellipitic Parallel Postulate. So this means that Spherical Geometry is based on the negation of the parallel postulate.

Therefore, both Spherical and Hyperbolic geometries was made possible by mathematicians negating the parallel postulate.

If you place a sphere on a three-dimensional plane and split the sphere into eight congruent parts which results in eight congruent spherical triangles being created. These congruent triangles have three congruent angles whose measures are ninety degrees. Therefore, the angles of the triangle have a sum of 270 degrees for each triangle.

The fifth postulate states that if two lines in the same place...