I would think a ship would have to pass within 50 miles to see an island so it seems remarkable that most, if not all, of even the most isolated island groups were discovered by europeans by the end of the 18th century.
Was exploration really this voluminous?
The island Tristan da Cunha, 1500 miles from South Africa, 1200 from Saint Helena was discovered in 1506.
Saint Helena, also one of the most remote islands in the world, was discovered in 1502.
Bermuda, 665 miles from North America, discovered in 1503.
Magellan discovered guam and the Mariana Islands in 1521.
South Keeling/Cocos Islands were discovered in 1609.
the Marquesas Islands were discovered in 1595.
I'm guessing that in some situation, the local population on some islands could have pointed in them general direction of other islands but it still seems like there must have been a lot of luck involved.
A reply to /u/MrEngleman
What a good question!
During the early part of the age of exploration, each voyage is a mix of exploration and exploitation or trade, in varying degrees. Luck certainly played a big part in the discovery of those islands you listed, but to those examples I raise the Hawaii islands, who weren't definitively discovered until the 1770s, after centuries of European navigation across the Atlantic.
Voyages such as that of Columbus and Magellan were firstly voyages of exploration. After Columbus made his way to Hispaniola, several other Spanish groups explored around there, and Columbus himself either led or sent expeditions to map the areas around Hispaniola. As the route between Hispaniola and Spain was starting to grow in traffic, Bermuda was discovered during a return voyage by one of the captains.
Tristan de Cunha may have appeared distinctly remote, but fortunately for history it lies near a preferred route for the Portuguese as they traveled to Asia. They tended to sail SW to Brazil, continue SSE to the Roaring Forties, and then sail east toward South Africa. Tristao da Cunha discovered that island during such a voyage. St. Helena was discovered in similar fashion.
Magellan's discovery of those islands were a mix of luck and tenacity. He was specifically contracted by the King of Spain to undertake an exploration to find a westward route to the Indies. This was in an era when N-S measurement was available but W-E was not. The Europeans already knew the N-S position of the Indies, and an idea of the size of the earth, but it was still difficult to ascertain the speed of the vessels. That's why if you look at the route that Magellan took, it appears as though he overshot to the north a bit. Throughout, he specifically looked for land masses and explored them when he could.
So in summary, you should look at each of those discoveries individually and place each of them within the correct context. But for each discovery there had to be an element of luck involved, and the context increases the likelihood of striking land.
FWIW all of theses places were actually discovered sometimes thousands of years before those dates.