I'd beware of using secondary sources (Atlases, Encylopedias, etc.) to draw conclusions about the geography of Arda. The ones by David Day, especially, are quite inaccurate.
There is only once set of maps of Arda as a whole that might possibly be considered authorative, and those are to be found in The Ambarkanta, The Shaping of Middle Earth (HoME IV). The most important and probably the most accurate of these is map V. It shows the whole of Arda in the as it was in the First Age. For those who don't have access to the HoME books, The Enclyclopedia of Arda have a (mostly) faithful reproduction of map V, under Arda.
Tolkien envisaged his mythology to take place in
our world, and this is clearly evident from even a cursory glance at the map. Middle Earth itself is Europe, and possibly some of Asia; Africa is also clearly visable, along with a rather odd continent called the 'Dark Lands' which might, in position if not in shape, resemble Australia.
It is difficult to know what level of accuracy to ascribe to this map, but it is really all we have to go on. It belongs to an early phase of Tolkien's mythology and so taking it along with the later Silmarillion and The Lord of the Rings is questionable. It's a shame that the Professor didn't provide anything more detailed later in his life.
The original question concerned continents in the Third age. For this all we have to go on is map V and information about how the world changed with the drowning of Numenor. The world was made round, and Valinor and Tol Eressea removed. The creation of 'new lands' at this time is refered to in the Alkalabeth, which might possibly explain the abscence of the Americans from map V. However, this is speculation at best.
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...similar but different on account of things like continental drift...
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I hardly think so: continental drift didn't become an accepted scentific theory until the 1960s. The map I refered to above was drawn in the late 1930s.