The initial discovery voyage would probably have been considerably longer than 35 days. Once they left Tahiti, I imagine they did some serious wandering around before they came upon Hawaii. The taste of the water flowing endlessly down the slopes in Hawaii must have been incredible after having gone heaven knows how long barely surviving on the little rainfall they could capture in their ships.
That depends on the shape of the search pattern they usually used. I can imagine an exploring people using a radial-spoke pattern, for instance: Pick a random direction and sail straight out in that direction until you find something or your supplies are half-expended, and then come straight back. Using that strategy, you’ll usually find nothing, but occasionally find something as quickly as possible.
One article I read said that the best way to explore while ensuring you could still get back home would be wait until the prevailing winds were interrupted for a while, and then sail as far in the opposite direction until the winds reverted to their usual pattern. That way you would be brought back to the area you started from.
I think that makes sense if they had some idea there was something in that direction - I doubt they just randomly started out toward one of the horizons without some indication they would have a better than decent chance of finding an island. Unless one of the reasons for setting out in the first place was duress, such as losing a war, famine, or overcrowding. I suspect there was more thought put into these expeditions than mere chance.
Rapa Nui, tho, is a conundrum because it is so remote. Finding and navigating to many of the other islands in Polynesia seems plausible because of clues and evidence as discussed above, but Rapa Nui is so far from the next island it’s difficult to imagine what clues they may have followed to find it. Haystack, needle, and all that. In this case, perhaps a more likely scenario could be an expedition made a mistake and went out too far, or got blown off course into unknown waters, or whatever, and the island appears and saves them. I agree it is unclear if they had visitors from other islands given the distance, but clearly if there was any interaction it eventually ceased and the Rapa Nui people were on their own. Perhaps this island was found and inhabited by mere chance and not so much by planning.
One thing we will never know is how many expeditions departed never to return.
It is really interesting to think about how we humans were able to locate and occupy even the most remote specks of land surrounded by the biggest and least habitable environment on earth (for humans), by using our creativity, ingenuity, and willpower.
Regarding Rapa Nui, one theory is that they used sea turtles. Supposedly the turtles follow the same route every year. They would tie a migrating turtle to one of the canoes and then sail in the direction the turtle went. My source on this is an iFit walk of Rapa Nui with the trainer / guide Dr. Dale Simpson who is an archeologist specializing in the archaeology of Rapa Nui.
Turtles, sea birds - that was the point of the posts above. The Polynesians knew the clues that the wildlife gave them, so they did not have to be in visual distance to find an island. A couple of hundred miles from the island, observing what sort of birds, how many, which directions they flew would point them to available land. Ditto with turtle swimming directions. When the ocean is your livelihood and there’s no facebook or sitcoms to distract, humans can learn and memorize a lot of detail about the environment.
If finding land is the only purpose of exploration, then I can see the random-direction technique being considered too unprofitable. But there might also have been intangible benefits. Maybe going on a voyage of exploration is a rite of passage, or how young men prove their valor without having to go to war with the next tribe over. If that’s the case, then going on an expedition might be worthwhile even if you don’t find anything (though obviously, it’s much better if you do).
Others have posted about multiple methods for determining that one is near an island, or even that an island is ::points:: that way. But does anyone know how close one needs to be for these to work?
Assuming some sort of grid (I suspect actually a tacking zigzag) search, that would tell us the mesh size needed to find most of the islands that they ended up finding.
Most islands will develop a cloud which is puffy and white in the afternoons. if you are at sea you can see this cloud from mlies away, long before you see the the land it is standing over. The winds also carry the smell of land. In Bali, for example, winter brings the smell of eucalyptus on the South wind blowing up from Australia. Borneo’s “slash and burn” farming sends smoke into the islands around Java in the other season.
The Maori navigators named New Zealand “the Land of the Long White Cloud” for this reason.
Turtles are for whimps. Real men use sharks.
More specifically, the mountains of NZ produce lee waves and associated lenticular clouds that are notably long (sometimes hundreds of miles). These may form at high altitudes (as much as 30,000+ ft) and would thus be visible a very long way from land.
This has been one of the most interesting questions and answers in a long time. I know nothing about the topic so haven’t contributed, but both the question and the answers have been fascinating. 
The Polynesian people are a sea going race who learned to live on the ocean for long periods. The night sky was their map, and they routinely were blown far from where they meant to be. Finding a paradise like Hawaii would of course encourage them to return. Our reliance on modern technology can blind us to what the mind can do with an awareness of the environment and history passed down verbally. Europeans did not recognize the writing used by Mesoamericans and so destroyed written records spanning hundreds of years. Knots on a string may not convey meaning to me, but to another they could be a complete description of a battle. It was happenstance that Hawaii was found, but it was colonized purposefully.
Yes, Yakamaniac, we already knew that they did it by being clever and smart and well-practiced and so on. But that doesn’t answer the how.
I don’t know how far off you could see land from standing in a canoe. I imagine the islands with central mountains were the easier ones to see. But with variable winds and currents, the same skills that would let them find unknown islands would be necessary quite often to be sure they were on the correct course to travel between known islands hundreds of miles apart. Particularly, things like the mentioned bird life, cloud shadows, waves etc. would be a clue as to whether the canoe was say, missing land and how much to turn left or right to correct course.
I think people underestimate how long people can survive at sea if properly prepared. The survivors of the Essex which had been wrecked by a whale, were able to navigate almost 2,000 miles in 93 days. All they had for provisions was some bread and waterlogged rations and some Galapagos tortoises. Many of them starved to death on the journey but that is still a long way in small boats with little provisions. The Kon Tiki expedition traveled 4,300 miles and I don’t think they were experienced sailors. Ancient Polynesians would probably been able to set out on journeys of discovery that covered thousands of miles and lasted months and come home safely if they did not find anything.
They were all young men (and fearless), and it was soon after WWII (so going out and doing something and getting killed while doing it seemed an ordinary kind of activity to them).
Something similar may have been true about early Polynesian explorers.
Captain Bligh and 18 loyal crewmen were put into a 23-foot open boat by the Bounty mutineers, and traveled over 4,000 miles in 47 days to Timor, the nearest European settlement, with very little food. They were prevented from landing on most islands on the way by hostile natives. Bligh lost only one crewman, who was killed in a native attack.