geometry nevertheless. One model for it is the geometry on the surface of a
sphere. Here the great circles (those circles that have the same circumference as
the sphere itself) play the role of straight lines in Euclidean geometry. In this
non-Euclidean geometry the sum of the angles in a triangle is greater than 180
degrees. This is called elliptic geometry and is associated with the German
mathematician Benhard Riemann who investigated it in the 1850s.
The geometry of Euclid which had been thought to be the one true geometry
- according to Immanuel Kant, the geometry ‘hard-wired into man’ – had been
knocked off its pedestal. Euclidean geometry was now one of many systems,
sandwiched between hyperbolic and elliptic geometry. The different versions
were unified under one umbrella by Felix Klein in 1872. The advent of non-
Euclidean geometry was an earth-shaking event in mathematics and paved the
way to the geometry of Einstein’s general relativity (see page 192). It is the
general theory of relativity which demands a new kind of geometry – the
geometry of curved space–time, or Riemannian geometry. It was this non-
Euclidean geometry which now explained why things fall down, and not Newton’s
attractive gravitational force between objects. The presence of massive objects in
space, like the Earth and the Sun cause space–time to be curved. A marble on a
sheet of thin rubber will cause a small indentation, but try placing a bowling ball
on it and a great warp will result.
This curvature measured by Riemannian geometry predicts how light beams
bend in the presence of massive space objects. Ordinary Euclidean space, with
time as an independent component, will not suffice for general relativity. One
reason is that Euclidean space is flat – there is no curvature. Think of a sheet of
paper lying on a table; we can say that at any point on the paper the curvature is
zero. Underlying Riemannian space–time is a concept of curvature which varies
continuously – just as the curvature of a rumpled piece of cloth varies from point
to point. It’s like looking in a bendy fairground mirror – the image you see
depends on where you look in the mirror.
No wonder that Gauss was so impressed by young Riemann in the 1850s and
even suggested then that the ‘metaphysics’ of space would be revolutionized by
his insights.