What is the rising action of Carry On, Mr. Bowditch?
The climax is Bowditch's navigational feat of safe journey to and return from Sumatra with a ship load of pepper. Everything that comes after the exposition but before the climax is the rising action.
Aside from the action where, during his time as an indenture servant, Bowditch learns everything he can about all subjects that circumstances introduce to him, the most important rising action occurs when he is at sea; when he learns of the ship wrecks caused by errors in Practical Navigator written by John Moore; when he decides that in order to save lives he must himself write an error-free, mathematically correct navigational book.
Nathaniel Bowditch completed his indenture and took employment as a surveyor, a profession in which he could use his exceptional mathematical skills. Soon he chose to go to sea instead.
During the journey to the Isle of Bourbon, he both used his navigational skills (another professional skill dependent upon exceptional mathematical ability) and taught navigation to the seamen, who were delighted to learn and who felt pride in themselves for their great accomplishment.
"According to my figures, sir, we're sixty-one miles east of our dead reckoning."
Captain Prince shook his head. "We couldn't have overrun our reckoning that much!"
"If my figures are right, sir, at our present speed, we'll be sight Bourbon on the eight."
It was [on] the morning of the eighth when he heard the lookout's singsong, "Land, ho-o-o-o-o-o!"
Captain Prince ...swept Nat with a sidelong glance. "I believe you can work a lunar, Mr Bowditch."
... "It's a simple matter of mathematics, sir."
To his dismay, Nathaniel discovered errors in Moore's mathematical calculations in Practical Navigator. He was dismayed because errors in navigational charts meant lives needlessly lost at sea.
In between sea journeys, Nathaniel married Elizabeth Boardman, who, sadly, died while Nathaniel was away at sea. During his period of grief over his wife's death, a daring detour to Manilla during a monsoon called attention to Nathaniel's skillful navigation, based, of course, on meticulously correct mathematical calculations.
News of further deaths--occurring during sea voyages and due to errors in Practical Navigator--awaited him, Lem (later found alive) and his brothers, William and Hab. With so many needless, navigation related deaths surrounding him, Nathaniel determined to work night and day and produce a completely accurate, error-free navigational guide.
At the climax of the story, when New American Practical Navigator was completed and published, he advertised it to the sea-faring world by using it to navigate his own ship, captained by himself, successfully to and from Sumatra, Indonesia, after picking up a valuable load of Indonesian pepper.