According to The Silmarillion, Arda (Earth) was originally flat; when Numenor tried to invade Aman, Eru Iluvatar reshaped the planet. He made Arda spherical, and set Aman apart as a separate "planet" (for lack of a better word). But a flat planet raises some interesting problems:
Assuming that the sun works the same way as it does with spherical planets (i.e., illuminating the entire side of the planet facing towards the sun), each of the two sides of the planet would experience night for about 12 hours, then day for about 12 hours.
Assuming further that one side contains all the oceans and continents and the other side is basically empty and uninhabited, everyone and everything on the good side would experience sunrise and sunset almost simultaneously (as opposed to the way things work for us, where a boundary between day and night is constantly advancing across the planet's face, and dawn in New York happens at the same time as dusk in Tokyo).
You can see what I mean by imagining what would happen if you took a table, pointed the beam of a spotlight at it, then raised the spotlight from a position below the table to a position above one end of the table. Instead of "daylight" slowly spreading from one end of the tabletop to the other, the whole tabletop would suddenly jump from "night" to "day".
Arda wasn't as smooth as a tabletop, of course - it had mountains and valleys and highlands and lowlands, so some areas would linger in shadows until the sun rose above the raised obstacles (mountains, hills, highlands, etc.). Still, for the most part, on a flat planet, dawn would not last quite as long, and would occur almost simultaneously across the entire face of the planet. The same is true of dusk/sunset - it would be a bit briefer, and would almost simultaneously occur across the whole face of the planet.
This would be like dawn happening at the same exact time in New York, Paris, Moscow, Beijing, Melbourne, Los Angeles, Chicago, and everywhere else on earth. The world would spend about 12 hours enjoying daylight, then sunset would reach everywhere simultaneously, and everyone would spend about 12 hours in the dark.
There would be other problems, too: If Arda didn't wobble like our spherical earth does (this is called "Axial tilt"), it wouldn't have seasons. The planet would only receive direct sunlight (roughly perpendicular to the planet's surface) for a few minutes around noon each day. This would prevent plants from establishing annual growth cycles, and interfere with animal migration and hibernation. The planet would have warm days like our summers, but relatively short days like our winters. If Arda wobbled like earth, the entire planet would experience the same seasons at the same times, as opposed to our system, where summer in the northern hemisphere is winter in the southern hemisphere. This would wreak havoc on weather patterns and make all the seasons more intense.
There are "flat earthers", who insist that we live on a flat planet. They account for half the planet being dark while the other half is bright by claiming that the sun is closer to the earth, and much smaller, than scientists suggest; they believe it casts a relatively narrow beam that wanders across the face of the planet in a cyclical pattern, illuminating one area, then another.
A small, nearby sun wouldn't work this way on a spherical planet, though, so even if Arda originally conformed to the flat earther system, Eru would have needed to change the sun (making it bigger, brighter, hotter, and more distant) when he made Arda spherical.
And so we come, at long last, to the point:
What was Arda like before Eru made it a sphere? Did it have no real seasons, simultaneous sunrises /sunsets across the whole planet, and consistent, unvarying 12 hour days and 12 hour nights? Was the weather roughly similar to ours?
These issues can be collected into a single question: Did Eru change the entire universe after destroying Numenor, or did he merely change Arda and Aman?
Obviously, Tolkien was an author and a philologist, not a physicist or an astronomer. But these problems would have been as apparent to his audience in the 50's as they are to us today. Surely someone would have brought up this question in a letter, at least?
Note 1: I have only addressed a few of the problems inherent in a flat-planet scenario. I have ignored others, including the fact that gravity would pull you towards the center of your side of the planet. Walking towards the edge would feel like walking up an increasingly steep slope as you approached the edge, and you would feel an ever-increasing danger of falling backwards away from the edge. Such problems are equally troubling, but not strictly related to my question here.
Note 2: I may have underestimated Tolkien. This site, which deals with the problems I'm raising, more or less, says the following:
In his later life, Tolkien began to have misgivings about the “flat earth” cosmology of the First and Second Age, and made outlines of a plan to create an alternate “round world” version of his legendarium in which the world had always been round. He never completed this revision before his death, and so the “flat earth” stories were published in The Silmarillion. Tolkien’s misgivings were primarily astronomical, having to do with the scientific absurdity of a sun that orbits the earth, a moon that glows from its own light, and stars that are points set in a firmament over the earth. How could our world emerge from such a beginning?
Tolkien Gateway even has an entry on the "Round World Version of The Silmarillion", which reveals how much the Flat Arda concept came to trouble Tolkien in his later life:
The Round World version is one of the variants of J.R.R. Tolkien's Legendarium, published in the final volumes of The History of Middle-earth. In this version, the setting of his legendarium is more realistic and less mythological: the Earth was always round, and Arda was the name for the whole solar system instead of just the Earth.
In the Round World version the Sun and the Moon were not the fruit of the Two Trees, but actually preceded their creation. The significance of the Trees and the Silmarils was that they preserved the light of the Sun before it was tainted by Melkor when he ravished Arien.
Similarly, the stars were not created with the Awakening of the Elves, but the occulting clouds were removed to reveal them, and it wasn't Varda who kindled them, since her power was limited to Arda while the stars were set in Eä.
This version emerged in writings from 1958-1960, but it was never developed beyond the stage of drafting and Tolkien didn't continue the revisions. Thus the Flat World version was chosen by Christopher Tolkien for the published The Silmarillion. Tolkien had previously attempted to write a round world version of the Ainulindalë and the Fall of Númenor, but in both cases he returned to the flat-earth model. Beside this, references to the seas being first "bent" after the Fall of Númenor, to the "Sunless Years", and to the trolls of the Twilight, survived in Lord of the Rings.
The Round World version can be deemed by Tolkienists as the definite 'actual' story behind the text; the text of the Quenta Silmarillion then, can be seen as just the legends based on the 'reality', written by the ancient people of Middle-earth. In his last years, Tolkien didn't view his legendarium as having an Elvish origin, but a Mannish one, and thus the legends contained in it could be inaccurate. This can be seen as a commitment to retain the older legends in the context of Mannish transmission, without need to rewrite the tales, as Tolkien had attempted at first.