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After Jim has breakfast, Squire Trelawney gives him a job: he is to deliver a note to Long John Silver at the Spy-glass inn. (A spy-glass is a telescope, by the way.)
Jim finds the inn easily. He feels shy about going inside, though: it's noisy and filled with sailors.
Soon Jim sees a man coming out of the side room: a friendly-looking fellow with one leg cut off close to the hip, moving around skillfully on a crutch.
Jim had been afraid that Long John Silver might be the one-legged man his captain was so afraid of back at the Admiral Benbow Inn. But once Jim sees this nice, open-looking fellow, he's sure the man can't be a pirate. After all, he's an innkeeper – how piratical can he be?
He goes up to the one-legged man and asks if he is "Mr. Silver."
It is, indeed, Long John Silver.
When Long John Silver sees Squire Trelawney's seal on Jim's note, he realizes he must be the Squire's new cabin boy.
Long John Silver shakes Jim's hand.
Just then a customer hurries out the side door. It's Black Dog, the man missing two figures who came to harass the captain along with Blind Pew.
Jim tells Long John Silver that the man, Black Dog, is a pirate; Long John Silver agrees that he hasn't paid his bill.
So Long John Silver sends a couple of guys after Black Dog to track him down.
Long John Silver asks Tom Morgan, Black Dog's drinking buddy, if he had ever seen Black Dog before.
Tom Morgan swears that he didn't even know his name.
Long John Silver asks what Black Dog had been talking about.
"Keel-hauling" is the answer.
Keel-hauling is a truly awful punishment supposedly popular among pirates – actually, it was much more popular among the Dutch and British navies way back in the day. The keel is the central beam that provides the spine of a ship. If you're unlucky enough to get keel-hauled, what happens is this: someone (possibly pirates) will attach you to a long rope that runs underneath the ship. Then you'll get tossed overboard. The rope will carry you quickly to the bottom of the boat. Because ships spend so much time at sea, they often collect barnacles and other sharp-shelled fish around the hull (the body of a ship). So as you sink under the hull, you'll get all cut up on the shellfish. Worst of all, if you're not pulled back up again quickly, you'll drown. The reason it's called keel-hauling is because you're literally going under the keel – the beam that anchors the two sides of the ship's hull together. (link)
Long John Silver tells Tom Morgan keel-hauling is a proper topic of conversation for such as him, and sends Morgan away.
Long John Silver claims that he has seen this Black Dog before – with a blind man.
Jim chimes in that this is Pew.
Long John Silver agrees that Pew is his name. He promises to run down Black Dog and keel-haul him himself.
Long John Silver decides to go tell Squire Trelawney all of this business himself.
Hearing the story of the appearance of Black Dog, Doctor Livesey and Squire Trelawney both agree that there is nothing else Long John Silver could have done.
They agree to set sail at four that afternoon.
After Long John Silver leaves, Doctor Livesey agrees that he seems like an honest man.