Topics in this math help tutor video: math history, What is the Truth of the Parallel Postulate without making any other
pure assumptions? What are Euclidean and non-Euclidean geometries?
On a sphere, the sum of the angles of a triangle is not equal to 180 degrees. The surface of a sphere is not a Euclidean
space, but locally the laws of the Euclidean geometry are good approximations. In a small triangle on the face of the earth,
the sum of the angles is very nearly 180 degrees.
During the eighteenth and nineteenth centuries mathematicians were increasingly questioning the foundations of geometry. This programme
shows how these investigations led to the formation of non-Euclidean geometry.
Also note that the 1970s public, for which this was intended, were apparently considered much smarter than people today! How could we allow
such a circumstance to develope?
Non-Euclidean Geometry is important to understanding Riemann curvature of space-time described by General Relativity. What was once considered
to be a fundamental truth of mathematics and reality, may now recognized as confusing the mathematics with a choice, or lack thereof, imposed
by nature, of coordinate system or perspective. That is, "may change of coordinate system result in a different, yet compatible in terms
of perspective, descritption of the laws of physics?"