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He proves this by admitting to the Captain that he has been on the island before. He claims he was there as a cook on trading ship that was forced to stop on the island for water. He claims that during the stopover he learned the pirates' names for places on the island and offers to help the captain find the best place to anchor.
He claims that the best place is an islet denoted as Skeleton Island, and that the previously identified highest hill is called Spy-glass, since it was the pirate's lookout. Because of his knowledge Captain Smollet asks Silver to look at a chart and identify the place the ship should anchor. All of this looks to Silver's ability to manipulate situations. By offering information, he's able to get a look at the treasure map... and he gets the information he needs with the captain's permission.
Treasure Island/ Chapter 12