Riemann space

[Monte Davis]

> So Riemann had the idea of the curved space before Einstein?

The idea of "a" curved space circulated by the 1820s in Bolyai and
Lobachevsky (Gauss had played with it  earlier but not published). Riemann's
1854 _Habilitationsschrift_ *generalized* their results, by showing that
non-Euclidean geometries could not only meet all the traditional
expectatiions of internal consistency, but support analysis -- calculus and
its descendants -- just as well as 3-space.

The idea that *this* space we live in is curved took longer to mature. AtD
portrays a period when it was one of the most interesting topics for
mathematics and the leading edge of mathematical physics (Klein, Minkowski).
But the real clincher was Einstein's GR in 1915: the first example of major
physics that *demanded* the assumption of curved space, yielding results you
couldn't get without it. Other AtD math tropes (quaternions, vector
analysis, imaginary numbers) had already proved their "applied value" in
electrical engineering. I'd say TRP cheats just a bit by inviting us to
project backward a decade or two what we now know the importance of the
Riemann-> Einstein branch would turn out to be. 

(Quibble #2: The Iceland spar (birefringence) metaphor doesn't have any deep
connection I can see with the other math in AtD; you can explain the dual
refraction perfectly weill with "early classical" math & physics, no Riemann
or quaternions or imaginaries required.)

> Did he have concrete geometrical figures in mind as images to describe his

notion of space?

For a few images, go to Wikipedia's
http://en.wikipedia.org/wiki/Non-Euclidean_geometry
 and then to 

http://en.wikipedia.org/wiki/Hyperbolic_geometry

Very roughly, if Eudlidean 3-space is "flat" (parallel lines stay parallel
forever)

...then elliptic geometry is curved "all the same way" like the surface of a
sphere (longitude lines that are paralllel at the equator converge at the
poles)

--- and hyperbolic geometry is curved "two ways" like the surface of a
saddle (there are two qualitatively different types of parallelism)