Dr Alien, on 30 October 2009 - 06:56 AM, said:
how the hell would they know how it looked like

Well, to be fair, it's hardly an accurate outline of America, either North or South. If you look at enlarged versions of the map, you can tell which coastlines have been visited by Europeans by the density of names along the land-side of the coast. It's particularly noticeable around Africa, and in a couple of sections of the east coast of North and South America. But all along the west coast of North and South America there are no names.
As for the existence of the Pacific Ocean, my suspicion is that it was an inspired guess by Waldseemueller which happened to be right. My logic works as follows:
1. Columbus was convinced he could sail west from Europe to China.
2. Jesuit astronomers (at least, I think it was them) said he couldn't possibly do so, based on their (fairly accurate) calculations of the circumference of the Earth and the width of Eurasia.
3. Columbus sailed anyway, and found land. He assumed this land was China.
4. If Waldseemueller had access to calculations of the size of the Earth, the approximate width of Eurasia, the approximate distance Columbus had travelled, and descriptions of what Columbus and others had found, he may have concluded that the land Columbus had found wasn't China. And seeing as this was the east coast of a land which wasn't China, then the land had to have a west coast facing the east coast of China.
I think the map is impressive, and if my theory is right, then Waldseemueller should be congratulated for his logic stream. But if my theory is right, it was still a guess, just a correct one. It might be interesting to have a look at other maps of the time to see what wrong guess other map makers made. Anyway, he still made mistakes - note that the islands off the coast of Asia are all wrong, and he has a gap between North and South America.