The Nautilus starts cruising the seas again. Aronnax tells us that they've covered 7,500 leagues. They're headed for the Persian Gulf now.
Ned's confused. The Persian Gulf is a dead end. They won't be getting to Europe anytime soon, even if they go through the Red Sea.
(A historical note: The Suez Canal, which connects the Red Sea with the Mediterranean Sea via the isthmus of Suez, was in the process of being built by this French dude, Ferdinand de Lesseps, at the time, but it wouldn't be finished until 1869.)
Ned, per usual, wants off the sub. Aronnax tells him there's nothing to do but wait.
After tooling around the coasts of Arabia for a while, Nemo steers the Nautilus toward the Red Sea.
At this point, Aronnax is sure Nemo's gonna pull a u-turn. He's got nowhere to go… but he keeps going anyway, and soon they're in the Red Sea.
They spend some time sub-ing around the Red Sea, pulling in close to the coast of Africa.
Aronnax and Conseil take some time to admire the life in the sea.
After a couple of days there, Nemo visits Aronnax and asks how he's enjoying things.
Aronnax praises the Nautilus. The Red Sea is notoriously difficult to navigate, but, as usual, Nemo's sub has had no trouble. "[Y]our ship is a century ahead of its time," he tells Nemo, "or perhaps several. What a shame that such a secret must die with its inventor!" (2.4.49).
Nemo gives no reply; he just changes the subject. They talk of the problems of navigating in the Red Sea and the origin of its name, of the Suez Canal, and of the history of other canals between the Red Sea and the Mediterranean.
Then Nemo drops a bomb on Aronnax: we'll be in the Mediterranean tomorrow, he says.
Aronnax can't believe it. There's no way they could make it all the way around Africa in so short a time, and there's no shortcut for the Nautilus.
The secret, Nemo tells him, is not going around Africa or over land, but under the water, through a secret passage, a passage Nemo has named the Arabian Tunnel.
See, Nemo discovered its existence after some careful observation and reasoning. He noticed that some of the same species of fish showed up in the Red Sea and the Mediterranean, and realized there must be some connection between the bodies of water. So he found the Arabian Tunnel.
Aronnax is, like, so totally amazed by this revelation.