New World/India

At what point in history did Europeans first realize that the Americas were a separate landmass and not connected to India.

The first European map labeling them as “America” and showing them as separate from Asia was in 1507:

Note also that Magellan’s expedition circumnavigated the globe in 1519-22. Columbus may have initially believed he had reached Asia, but it didn’t take long for the Europeans to figure out that it was the “New World”.

By tradition, Amerigo Vespucci is credited with this insight (the myth that claims that Vespucci never travelled to America has been debunked, incidentally - he did undertake voyages there). He might not actually have been the first person to realise this; in any case, it was worked out quite fast - after the initial discovery in 1492, it took Europeans only a few years at most to figure out that America was a new continent.

The only reason Columbus even thought he had reached India was because he was ignorant of information which was well-known for two millenia. The Greek mathematician, Eratosthenes (276 BCE-195 BCE) had proved that the Earth was round, and drived a figure for its diameter that was within 1% of the currently-accepted value.

As it was, on Columbus’ first voyage, the trip took longer than expected and the crew became restless, when they spotted land, far out in the Atlantic from the American continents. Had the American continents not been there, they would have had to traverse open ocean the width of North America, and an entire hemisphere of water over 10,000 miles!

For all the hype about Columbus, his voyage proved to be a spectacularly poorly-researched and ill-concieved leap into the void, and it is only by a staggering stroke of luck that he and his crew did not simply disappear, never to be heard from again.

Not quite. The idea that North America might be connected to Asia – essentially, a hitherto unknown extension of Asia – persisted; I believe that there were maps attempting to make this idea work right into the mid-1500’s. (Of course, except for the Bering Strait, North America practically is an extension of Asia.)

(I read a book about this exact topic recently – unfortunately I can’t remember the title or author.)

Columbus did not think he was in India, he thought he had landed near Cipango (Japan). His desire was to go to Cathay (China). He had trouble convincing anyone of the merits of his expedition because of two foolish errors. First, he overestimated the distance from Europe to Cathay traveling East. Second, even though the diameter of the Earth was known quite accurately, he believed it was smaller than the accepted value. Measuring the diameter of the Earth is not difficult, so this was his biggest mistake. Without these mistakes, of course, none of us would know his name.

When Columbus headed out on his first expedition was it well known that he was going and where he was headed? Certainly the crew must have known, but was it common knowledge in Spain at least? Or was his mission kept secret from the public? Or was he considered a crank who would likely die trying to sail across half the world?

Found it! Terra Cognita: The Mental Discovery of America by Eviatar Zerubavel.

“India” as a term was much more loosely and widely applied then than it is now. It did not refer just to the subcontinent or the present-day Republic of India, but to much of Southern and Eastern Asia. In that sense, Columbus did think he was in India.