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Hornblower and the rest of the men under his command are recaptured by the French after the ship they've commandeered sinks. The Indefatigable is sighted on the horizon, but the French ship is much faster. So what does Hornblower do? He lights the ship on fire. And then refuses to take credit for it as a self-inflicted punishment for losing the French ship.
Volunteering to help Spaniards save their men, then not going back with the British. To make up for it, he is set free by an order personally signed by the Prime Minister of Spain.
Beat To Quarters/The happy return: Both encounters with the Natividad.
First time, he takes the ship in a surprise boarding attack at night. The Spanish/native crew, who hasn't seen a serious fight in decades, is overwhelmed in minutes.
Second time, an open battle, spread over two days, between the 36-gun Lydia and the 50-gun Natividad. After the first round, both ships, severely damaged, drift apart. Within one day, Hornblower has his ship fit for battle again and by anticipating his opponent's thinking, manages to find the other ship again. Then, as the Lydia has to be pulled into battle by her boats in a dead calm and under fire, he has his crew dance the hornpipe to keep morale up. Once they are close enough to return fire, the ''Natividad'' is sunk within an hour.
Ship of the Line: Hornblower's Heroic Sacrifice of his ship. He manages to cripple three out of the four French ships before he had to surrender (and the fourth ship had to keeps its pumps running round the clock following the battle).
Earlier, Hornblower having the Sutherland tow the flagship Pluto out of danger, in the middle of a storm that had just dismasted the Pluto.
Flying Colours: Hornblower, Brown, and Bush's escape from the French. In the snow. At night. And Bush is an invalid at the time. Later they sneak out of France by recapturing a British ship from the French.
Also, Hornblower's trial. Under the Articles of War, a captain who lost his ship was automatically to be tried for not having done his utmost best to prevent her loss, and Hornblower had lost the seventy-four gun ship of the line HMS Sutherland in Ship of the Line. Between the circumstances of the loss and him returning with a recaptured prize and a group of sailor to be pressed into service, the trial is just an excuse to make him a war hero and have him honourably aquitted.
Commodore has too many candidates to count, as does Beat to Quarters.
Many in Lieutenant Hornblower. Of note are Hornblower's ability to maneuver his superiors magnificently—after the Captain Sawyer's accidental-or-not fall, he convinces the indecisive and timid Buckland to open the secret orders and attack the Spanish fort. When the first attempt fails terribly and he's called into a war council in the middle of the night, he configures a second plan within minutes; it succeeds brilliantly and the fort is taken with hardly any British casualties. Once there, he figures out how to heat cannonballs without any prior experience and then devises a way to put a ship's cannon up a cliff to threaten the escaping Spanish ships, which results in their unconditional surrender. And, to round it all off, leading the boarding party to recapture the Renown after it has been taken over by the Spanish prisoners.
Hotspur: The chase with the frigate Loire. Hornblower manages to pull off more than one Batman Gambit against the French captain without ever having met the man—all he has to do is observe the Loire's maneuvers. He even puts the huge warship in a defenseless position at one point so that his little sloop can fire a broadside unopposed!
The Tear Jerking moment when Archie sacrifices everything, mainly his good name and honour, to save Horatio from being executed.
Captain Sawyer and the boy who he had repeatedly beaten near to death stand united at last and make one final stand.
Captain Pellew shooting Jack Simpson, who, apparently not satisfied at having cheated during his duel with Hornblower, was attempting to stab the latter in the back. Pellew's coolness as he fires and Horatio's stunned and impressed expression make it that much more awesome.
Bowles: Exceptionally fine shot, if I may say so, sir. Pellew: You may, Mr Bowles. You may.
"The Duchess and the Devil": Hornblower and co rescuing the Spanish sailors from a reef. In the middle of a storm—then returning to prison even after having been rescued, all because Hornblower gave his word to the prison warden. Luckily, Spaniards are so grateful and impressed that they are soon granted freedom.
In "Even Chance", Horatio, in charge of a captured French ship, rescues the Indefatigable from three French corvettes by not running down the French colors. And then one of his shots hits a corvette's powder magazine. Pellew actually feels sorry for the enemy.
In "The Duel/The Even Chance", Horatio tossing the compass over the side of the jollyboat after the Marie Galante has sunk, the jollyboat taken by the French, and a gun pointed at him by their captain. It's treated as an in-universe one by the sailors once they get back to the Indy and immediately start recounting the moment.
Horatio: Fish for it.
Archie defending Wellard after Randall mocks his authority, reminding everyone firstly that Wellard is in fact an officer and also that Archie will not let him get mistreated if he can possibly help it.
Hornblower's examination for lieutenant in "The Fire Ships" starting from the time the examining board takes note of the cannon shot. Climbing on a wooden ship full of fire and steering it away from his home ship more than makes up for his utter failure at the hypothetical problem posed to him in the actual exam.
Hornblower, who is afraid of heights, runs out along a yardarm forty feet above the deck to loose a sail. Bush, who can't swim, calmly hangs onto the ship's anchor as the rowboat sinks under him (and eventually with him). An officer in HMR Navy has a duty and phobias and survival instincts can all go to hell, thankyouverymuch.
A true Awesome Moment for Bush was when the Spanish prisoners escape and quietly get to work capturing the Renown. One of them sneaks into his room to find his bed empty... and Bush steps from behind the cabin door and shoots the Spaniard before sounding the alarm.
Simply the character of Majorly Awesome Major "My lord" Edrington, who, as one reviewer/snarky commentator put it, starts off an uptight aristocrat who thinks very little of Horatio and Archie and ends up an uptight aristocrat who thinks quite a lot of Horatio and Archie.
From "The Duchess and the Devil": Horatio's honest answer to Kitty Cobham that he wouldn't have entrusted her with his dispatches had he known she was no duchess, but that now when he got to know her, he trusts her completely.