What are the two main branches of NonEuclidean Geometry What

What are the two main branches of Non-Euclidean Geometry? What is the primary difference between them? Describe the historical development of Non-Euclidean Geometry, include work done by Lobachevsky, Playfair and Riemann.

Solution

Two main branches of Non-Euclidean Geometry-

a. Hyperbolic Geometry

b. Elliptical Geometry

Primary difference-

In hyperbolic geometry, the lines \"curve away\" from each other while in elliptical geometry, the \"lines curve\" towards each other.

History-

Non-Euclidean geometries were not widely accepted until the 19th century. The discovery of non-euclidean geometry began when Euclid\'s work \"Element\" was written. Euclid began with a limited number of assumptions and sought to prove all the other results in the work.The most notorious of the postulates is often referred to as \"Euclid\'s Fifth Postulate\" which tells about hyperbolic and elliptical geometry.

Lobachevsky booklet explains that-

All straight lines which in a plane go out from a point can, with reference to a given straight line in the same plane, be divided into two classes - into cutting and non-cutting. The boundary lines of the one and the other class of those lines will be called parallel to the given line.

Playfair explaination-

Given a line and a point not on the line, it is possible to draw exactly one line through the given point parallel to the line.

Riemann gave theory of spherical geometry. In the Riemann type of spherical geometry, lines have no (or more precisely two imaginary) infinitely distant points.