There's a long list of what have been nicknamed "phantom islands" -- islands that have been recorded on maps, sometimes for centuries, but then when people follow the map and go out where the island supposedly is, there's nothing there.
Well, there's something there, namely a shit tonne of salt water. In one way, it's unsurprising that misidentifications like this can happen; icebergs, pumice rafts from volcanic eruptions, and even low cloud banks in the distance can look like land, and when you couple that with the desperation to reach terra firma a lot of mariners felt after weeks at sea, it's understandable that this sometimes occurred. What's more curious is how persistent some of these phantom islands were -- there are ones that were only conclusively proven not to exist in the last two decades.
A big part of the problem is that in the days before satellites and GPS, when you were out at sea, it was awfully hard to be certain of exactly where you were. Latitude, as it turns out, is fairly easy; in the Northern Hemisphere, the altitude of Polaris above the horizon (which you can measure with a sextant) is equal to the latitude. (It's a little trickier in the Southern Hemisphere -- there is no "South Star" -- but with a little adjustment, the same principle can still be used.)
Longitude, on the other hand, is a whole other can of worms.
You can figure out your longitude using the rising times of various stars, but the hitch is that requires you have an accurate timepiece that isn't thrown off by the incessant jostling and jolting on board ship. It wasn't until the eighteenth century that such a clock was invented, and it only went into widespread use in the nineteenth -- how this happened is the topic of Dava Sobel's wonderful book Longitude -- but even with more accurate timekeeping, figuring out exactly where in the trackless oceans you were was no easy task. This is probably what happened with the nonexistent Saxemberg Island, first sighted in 1670, which appeared on maps for almost two hundred years (and was "viewed extensively from a distance" in 1804 and again in 1816). It's now surmised that they were actually seeing the remote Tristan da Cunha Island, and had simply miscalculated where they were.
One that is likely to have been a combination of inaccurate longitude calculation and seeing something that looked like an island but wasn't is "New South Greenland," which was "discovered" by the curious figure of Captain Benjamin Morrell, originally of New York. To say that Morrell had a checkered career is a bit of an understatement. He ran away to become a sailor at age seventeen, served during the War of 1812 (and was captured twice by the British), but eventually rose through the ranks to captain the Wasp, which he took down into Antarctic waters in 1823. He had a penchant for exaggeration and occasional outright lying, but in this particular case he seems to have simply been mistaken. He reported an extensive land which he initially thought was part of the Antarctic Peninsula, and sailed along it for five hundred kilometers -- but subsequently he found his position to be ten degrees of longitude (at that latitude, about two hundred kilometers) east of where he thought he was, in a part of the ocean that has no land masses whatsoever and by later sounding was found to be 1,500 meters deep. So what he saw clearly wasn't part of Antarctica. What it actually was remains a mystery -- the best guess is a long connected chunk of icebergs.
For what it's worth, Morrell's career didn't improve much thereafter. He was involved in piracy in China and Madagascar and was lucky to escape with his life, launched a fruitless search for gold in New Guinea, and supposedly died "of a fever" in Mozambique in 1838 -- although a letter with his signature showed up in New York in 1843, leading some people to believe he faked his own death to get away from all the people he'd defrauded or otherwise pissed off.
Sometimes imaginary islands get wrapped up in mythology, and that makes it even harder to tease out what's real and what isn't. Penglai, "thirty thousand leagues off the east coast of Shandong, China," described as one of the homes of the "Great Immortals," is pretty certainly a tall tale -- although interestingly, there's a legend both in Vietnam and Japan pinpointing an island in more or less the same place (where it's called Bồng Lai and Hōrai, respectively). Saint Brendan's Isle, supposedly first seen in 512 C.E. by the Irish monk/explorer Saint Brendan of Clonfert, is another one around which wild tales have arisen, but it was reported so persistently that its existence was considered a fact for hundreds of years. (Its reputation for being the home of devils and demons led a priest in the Canary Islands to perform an exorcism directed toward the entire island in 1723.) The last alleged sighting of Saint Brendan's Isle was in 1772, but it still appeared on maps -- somewhere off the west coast of Africa -- well into the nineteenth century.
Sometimes islands do exist -- temporarily. This seems to be the case with Bermeja, discovered by Spanish explorers off the coast of the Yucatán Peninsula in 1539, and extensively described (along with its precise location) by other mariners in the sixteenth century. Expeditions to find it later proved unsuccessful, although close to the reported location there is a significant seamount. It's likely that Bermeja was the victim of a combination of erosion and tectonic shifting, and what was once dry land now isn't.
A lot of them, though, have eluded explanation except as mirages. This is almost certainly the case with the aptly-if-unfortunately-named Fata Morgana Island (a fata morgana is a common type of mirage experienced at sea, especially in polar regions). The explorers Johan Peter Koch and Aage Bertelsen reported it -- once again, along with an exact location, off the northeast coast of Greenland -- in 1907, and its existence was confirmed from the air by Koch's son Lauge in 1933. Unfortunately for all three of them, there's no land there, just lots of extremely cold salt water. The sightings were undoubtedly a combination of mirages and wishful thinking.
In any case, our precision GPS systems, satellite photography, and (I hope) less tendency to fall for fanciful tall tales has improved our ability to discern between what's real and what's not. Although I have to say I'm kind of disappointed that Antillia isn't real. A favorite claim amongst the Spanish and the Portuguese until the sixteenth century, at which point their own explorers came back and reported that there wasn't anything where it allegedly was but a big blob of the Atlantic Ocean, Antillia supposedly had seven cities run like some utopian paradise, where everyone lived in harmony and there was no crime or violence, and its leaders were wise, kind, and benevolent. I don't know about you, but if that one is ever rediscovered, I'm buying a plane ticket.
