Barnes & Noble has an informative new review today of the recent Galois book Duel at Dawn.
It begins…
"In 1820, the Hungarian noble Farkas Bolyai wrote an impassioned cautionary letter to his son Janos:
'I know this way to the very end. I have traversed this bottomless night, which extinguished all light and joy in my life… It can deprive you of your leisure, your health, your peace of mind, and your entire happiness… I turned back when I saw that no man can reach the bottom of this night. I turned back unconsoled, pitying myself and all mankind. Learn from my example…'
Bolyai wasn't warning his son off gambling, or poetry, or a poorly chosen love affair. He was trying to keep him away from non-Euclidean geometry."
For a less dark view (obtained by simply redefining "non-Euclidean" in a more logical way*) see Non-Euclidean Blocks and Finite Geometry and Physical Space.
* Finite geometry is not Euclidean geometry— and is, therefore, non-Euclidean
in the strictest sense (though not according to popular usage), simply because
Euclidean geometry has infinitely many points, and a finite geometry does not.
(This more logical definition of "non-Euclidean" seems to be shared by
at least one other person.)
And some finite geometries are non-Euclidean in the popular-usage sense,
related to Euclid's parallel postulate.
The seven-point Fano plane has, for instance, been called
"a non-Euclidean geometry" not because it is finite
(though that reason would suffice), but because it has no parallel lines.
(See the finite geometry page at the Centre for the Mathematics
of Symmetry and Computation at the University of Western Australia.)