- Tintin and the Captain search for the pirate's treasure they learned about the last story ("The Secret of the Unicorn"), with the help of their eccentric, but lovable new friend, Professor Calculus
- After solving the secret message of the three parchments, Tintin & Haddock plan their expedition to the island where Sir Francis Haddock was maroon centuries ago. Which they believe is the location of Red Rackham's treasure. Before the day of the exploration, Tintin & Haddock are visited by a hard of hearing professor named Cuthbert Calculas who provides help in their exploration with a Shark submarine. Which neither Tintin or Haddock were interested in having on their voyage. However due to Calculas believing they said they'' accept his submarine, he brings it the following day for the expedition to the deserted island. Upon visiting the island and sea diving in the area where the Unicorn sank. Finding Red Rackham's treasure proved to be a needle in the hay stack. Will our heroes be able to find the lost treasure?—J LeGault
- In the previous adventure, The Secret of the Unicorn, Tintin and Captain Haddock discover three parchments revealing the location of the Unicorn, a 17th century navy ship commanded by Haddock's ancestor Sir Francis Haddock. The Unicorn was scuttled by Sir Francis while battling the pirate Red Rackham for his treasure. Tintin and Haddock believe that the pirate's treasure is in the remains of the sunken Unicorn.
Tintin and Captain Haddock hire a fishing trawler, the Sirius, to search for the treasure. As the crew prepare for the search, their plans are discovered and publicized by the press, forcing Tintin and Haddock to deal with numerous strangers claiming to be Red Rackham's descendants and insisting on a share of the treasure. They are quickly driven away by Haddock, who reminds them he is the descendant of the man who killed Red Rackham.
Another petitioner is Professor Cuthbert Calculus, an eccentric and hard-of-hearing inventor who offers the use of a special shark-shaped, electrically powered one-man submarine to help search for the sunken ship without being bothered by the numerous sharks in the area. The submarine can dive up to 900 feet, and a 2-hour oxygen supply and a speed of 6 knots. The treasure hunters turn him down and prepare to embark.
Before Tintin and the Captain clear the port, the two detectives Thomson and Thompson join the crew to protect their friends from the possible threat of the rival treasure hunters, the Bird brothers who have escaped custody. Shortly after departure, Tintin and Haddock discover that Calculus has stowed away on board. The professor has stashed the parts of his submarine in the hold, removing the Captain's crates of whiskey in the process. Despite initially threatening to throw Calculus into the hold on bread and water, Haddock grudgingly decides to keep him along for the trip.
Tintin and Captain Haddock reach the location stated in Sir Francis Haddock's parchments. Initially, the party cannot find anything at the coordinates (20°37'42"N 70°52'15"W, off the Mouchoir Bank), but then Tintin hypothesizes that Sir Francis Haddock used the Paris Meridian instead of Greenwich (which would yield 20°37'42"N 68°32'1"W, off the Navidad Bank).
Sure enough, the ship reaches an unknown and uninhabited island. As they come ashore to explore it, the Captain stubs his toe on a piece of wood protruding from the sand, which is excavated and turns out to be the remains of Sir Francis Haddock's jolly boat. As they penetrate into the interior of the island, they encounter numerous skulls, which Tintin deduces are the remains of the island's cannibalistic former inhabitants. There is also a magnificent pagan icon of Sir Francis, and numerous parrots that repeat the Haddockian argot, which an amused Tintin realizes has been passed down for generations.
Calculus's submarine proves useful in searching for the sunken Unicorn, while the actual examination of the wreck itself is performed with a hardhat diving suit. Thomson and Thompson soon begin to rue their decision to join the treasure hunt, because they are consigned to manning the air pumps supplying the diving suit when Tintin, and later the Captain, explore the wreck. While facing complications like shark attacks, they discover a cutlass, a gold bejeweled cross, a strongbox of old documents, the figurehead of the ship and, to Captain Haddock's delight, a large supply of vintage Jamaican rum.
Although the search is otherwise unproductive, the crew spots a large wooden cross on the island itself and Tintin believes that the reference in Sir Francis' parchments to "the Eagle's cross" could refer to it as the marker for the treasure's location. Upon coming to the cross the party begins to dig, but after a while Tintin realizes that they are following a false lead, considering that Sir Francis would not deliberately leave his treasure on an island he did not intend to return to, so they return to the Sirius.
Time passes. Although there are further dives to the wreck, they are unable to find the treasure itself and they go home disappointed. Calculus's examination of the documents from the retrieved strongbox allows him to determine that Sir Francis was the owner of the large estate of Marlinspike Hall, the former home of the Bird brothers. Upon this discovery, Tintin insists that Haddock must purchase the estate, which is up for auction. Calculus, who has received large sums of money from the government after a profitable sale of his submarine design, helps his friend acquire his family home.
After purchasing the Hall, Tintin and Captain Haddock explore the cellars of the main house. Amongst the Bird Brothers' cluttered antiques, they find a statue of Saint John holding a cross. Tintin suddenly shouts out, "The Eagle's cross!" as he remembers the Saint is called "The Eagle of Patmos". At the statue's feet is a globe. On it, Tintin locates the island where Sir Francis Haddock was marooned. He touches that point and discovers it to be a trigger button-the globe springs opens, and Red Rackham's treasure is found hidden inside.
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