An account of how axioms underpin geometry and how by changing one axiom we get an entirely different geometry.
This article outlines the underlying axioms of spherical geometry giving a simple proof that the sum of the angles of a triangle on the surface of a unit sphere is equal to pi plus the area of the triangle.
Use simple trigonometry to calculate the distance along the flight
path from London to Sydney.
Compare the distance between two points diametrically opposite
each other on a line of latitude by travelling around the line of
latitude in the first case and by travelling directly over the
nearest pole along a line of longitude in the other case.
Make this comparison for different lines of latitude and show
that the ratio of the distances tends to a limit as the lines of
latitude get nearer to the pole.