We all know that in Tolkien's legendarium, the world was created flat. Valinor lay on the western edge. The sun and the moon were created long after the world, and flew across the sky every day. In the Second Age, that changed - the world was made round at the fall of Numenor, and Valinor was removed from the sphere. In essence, our own Earth was created.
Except that's backwards.
The Lord of the Rings - the only work on Middle-earth to be published in Tolkien's lifetime (other than The Hobbit, and we'll come back to that) - features several references to the time before the rising of the sun - a time when, according to our understanding, the world was well and truly flat, with no natural light other than the stars.
The world was young, the mountains green,
No stain yet on the Moon was seen,
No words were laid on stream or stone
When Durin woke and walked alone.
We know that the Dwarves were created long before the Elves awoke, but put to sleep so that the Eldar would still be the Firstborn. We also know that they were awake before the sun rose - Eol of Nan Elmoth was friendly with them, and their halls in the Ered Luin were deep and ancient.
So the statement that 'no stain yet on the Moon was seen' makes no sense. Of course there were no stains on the Moon - it was thousands of years away from being created! Furthermore, the mountains shouldn't have been 'green' - the plant-life of Middle-earth was hibernating in anticipation of the Sun.
But okay, that's a Dwarven folk-song. Gimli wasn't around back then, and it's not like we have anyone who actually was to listen to. Is it?
Ere iron was found or tree was hewn,
When young was mountain under moon;
Ere ring was made, or wrought was woe,
It walked the forests long ago.
That's Gandalf singing - and he's a Maia, he was there. Before ever ironworking or even fire were discovered - and the dwarves are fond of both - the Ents were awake - 'when young was mountain under moon'.
It is unambiguous, then, that in the text of The Lord of the Rings, the Moon existed long before the Two Trees died. Since there is no other account of it being created, we have to assume it was part of the first creation - that it's an orbiting rock, and that strongly implies a spherical earth.
But that's not all. Several times during The Lord of the Rings, Legolas performs feats which should be impossible.
It seemed to Legolas, as he strained his farseeing eyes, that he caught a glint of white: far away perchance the sun twinkled on a pinnacle of the Tower of Guard. And further still, endlessly remote and yet a present threat, there was a tiny tongue of flame.
Legolas is standing in Edoras, in western Rohan, and seeing the fires of Orodruin in Mordor. That's a distance of some 400 miles. For a 7-foot Legolas at sea level on a spherical planet, that would require Orodruin to be over 1700 miles high. The only way to make the numbers work is to have both the summit of Orodruin and the city of Edoras at the same altitude as the peak of Mount Everest - and even then, it's only barely possible. Of course, in that situation, Legolas would have other things to worry about, such as the lack of oxygen.
(Horizon distance is calculated as d ~ 1.22 x sqrt(h), where d is in miles and h in feet. At 29,000 feet, this distance is 207 miles. Observers atop Everest!Edoras and Everest!Orodruin would share line of sight on a point somewhere in the Nindalf, and thus would be able to see each other in a straight line. Except that they would both be asphyxiating)
How can we explain this? Does elven sight see round corners? Is the world of Arda much larger than our Earth? Or is the world, in fact, flat, giving a near-infinite horizon distance? It seems the only possible conclusion.
So Iluvatar made a spherical world, with its own sun and moon. Then, when Numenor rebelled, the world was 'made round' - circular, flat - to ensure the descendents of Numenor didn't get any more bright ideas... like, say, the notion of an orbit.
"Yes," said Legolas, "there are one hundred and five. Yellow is their hair, and bright are their spears. Their leader is very tall."
Aragorn smiled. "Keen are the eyes of the Elves," he said.
"Nay! The riders are little more than five leagues distant." - The Two Towers