VII - Double Elliptic Geometry

Abstract

This chapter discusses double elliptic geometry. In Euclidean and hyperbolic geometry, the existence of parallel lines is a consequence of the theorem that states that the exterior angle of a triangle exceeds each opposite interior angle, and this theorem is a consequence of the assumption that straight lines are infinite. Hence, a system of geometry whose straight lines are finite could not have any parallel lines at all. A system where any two straight lines meet in two points is called double elliptic geometry. It is also known as elliptic geometry of the spherical type because one is made aware of its existence by considering relations involving the entire sphere. The largest circles on a sphere are those whose radii equal the radius of the sphere. They are called great circles. The concept of a two-dimensional system of geometry that, such as Euclidean and hyperbolic geometry, employs the terms point, straight line, plane, straight line segment, distance, angle, congruent, perpendicular, and so forth, and whose points, straight lines, and straight line segments have the same properties and relations as the points, great circles, and geodesic arcs on a sphere, is the concept of double elliptic plane geometry.