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Answer by Stephen Tempest, qualified amateur historian:
Many islands went undiscovered until relatively modern times. In some cases, a mariner might report the presence of an island in a particular region, but subsequent voyages to the same part of the ocean might fail to find it again, and it might be several centuries until the island was rediscovered.
Still, there are ways to detect land when you're at sea that don't rely on modern technology.
If you're standing on the deck of a ship a meter above sea level, you can see a low-lying island 3.6 kilometers away. If your ship has a 30-meter-high mast and you climb to the top, however, you can see for 20 kilometers on a clear day.
On the other hand, the summit of Mauna Kea, the large mountain on Hawaii, is 4,205 meters above sea level. That means it can be seen from 232 kilometers away. In reality, that's unlikely to happen: Even in good weather, the air will usually be too hazy, and the mountain will be only a faint smudge poking above the horizon, easily missed or mistaken for a cloud. Even so, this shows that high ground can be seen for a surprising distance out at sea—especially if you know what you're looking for. Not all islands have high mountains at their center, though.
There are other ways to detect when an island is close: The easiest and best-known is by watching for birds. Birds need to nest on solid ground, but they might range out far over the sea in search of fish to eat. So if from the deck of your ship you see a flock of birds in the air, you know there's an island nearby. If you watch the birds for a while and see the direction they fly in, you can follow them and hopefully find the island they've come from.
There's another trick that is referenced in the Bible, in the tale of Noah's flood, and supposedly was also used by the Vikings during their exploration of the oceans. If you take a bird in a cage on the ship with you, then release it when you're at sea, the bird will fly up high; from this vantage point, it might be able to see an island in the distance that's too far away for you down at sea level to spot. If the bird flies off in that direction, follow it to reach land. If it returns to the ship, though, you know that there's no island in sight.
There are other ways to sense that land is nearby. The water may become shallower, resulting in different patterns of waves, or a different apparent color to the sea, or perhaps seaweed growing in the water. There might be driftwood. You might spot unusual weather patterns. For example, a mountainous island makes a windbreak on the otherwise flat sea, and this can cause clouds to form over the mountains. If you're sailing along under a blue sky and notice an unusual cluster of clouds just in one place, it's quite likely that there's an island underneath them. You might be able to hear the surf breaking onto rocks. You might even be able to smell the island from a distance, especially if it has an active volcano giving out lots of sulfur, but perhaps also if it's lush and tropical.
Early seafarers like the Polynesians or the Norse didn't have magical powers to detect islands, but they would, by necessity, be familiar with these signs of nearby land, and learn to trust them in a way that modern navigators with compasses and charts and GPS don't need to worry about. It should also be remembered that once previous explorers have reported that an island's there, it's much easier to reach it again: Sail for 15 days north-west, then look for land.
Even then, the island might be missed, especially if you reach its vicinity at night or in cloudy weather. You might sail straight past it, get lost at sea, and die of thirst in the open ocean. These things happened; long-distance sailing wasn't a safe profession.
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