In Weinberg’s textbook on General Relativity, he introduces the notion of intrinsic curvature by presenting a map of Middle Earth. Four points are marked on the map, with the six distances listed between them. He then asks whether Middle Earth is flat or spherical, and if spherical, what is the radius? (Four points and six distances is just enough to determine this, though not easily.)

Now, what I’m wondering is whether those distances correspond to anything from Tolkien’s works, published or unpublished, or if Weinberg just made them up. The distances listed are:
[ul]Hobbiton-Erebor: 813 mi
Hobbiton-Dagorlad: 960 mi
Hobbiton-City of the Corsairs: 1112 mi
Erebor-Dagorlad: 735 mi
Erebor-City of the Corsairs: 1498 mi
Dagorlad-City of the Corsairs: 780 mi[/ul]Can anyone find cites for these distances? Alternately, can anyone find some other set of four points in Middle Earth such that the distance between each pair of them is known? All distances must, of course, be as the crow flies, not by road.

That information would most likely be in or be derived from Karen Wynn Fonstad’s Atlas Of Middle Earth, not quite directly from Tolkien, but pretty close. At least the accuracy of the measurements could be established from that source. On a side note would the fact that Arda started out flat and later was made spherical affect the calculations at all?

Well, I don’t know anything worthwhile about geography or topology, but it is documented in Akallabêth that “Illuvatar showed forth his power, and he changed the fashion of the world” which had been flat throughout the First and Second Ages. In this cataclysm, the seas were bent. From this, it can be assumed that the world of the Third Age of Middle Earth was round.

Personally, I think it would be unimaginably cool if someone were to show that the maps Tolkien drew for the Silmarillion were shown to be of a Euclidian plane, while the map from The Lord of the Rings were of a sphere.