Nat Bowditch does not become a legend during the course of the book, but is remembered with great respect after his various adventures. Bowditch's main contribution to the maritime industries was his focus on practical and simple math to allow sailors and ocean-workers easier and safer navigation. His practical experience on the ocean under several captains and on several boats informed his work; Bowditch knew the importance of accurate mathematical calculations in regards to ocean navigation and so he worked to create formulas and equations which could be used by anyone to come to the same conclusions every time. This drastically improved the safety of sailing. In addition, Bowditch in the book is portrayed as an uncommonly-calm man, capable of instant mental calculations:

Nat stood near the compass, holding his watch so that he could read the face...
Watson in the larboard chains sang out, "By the deep ten."
From the starboard chains Kedzie yelled, "And a quarter less four!"
[...]
Nat's gaze never left his watch. He lifted one finger... "Now! Due west!"
(Latham, Carry On, Mr. Bowditch, Google Books)

In this scene, Bowditch uses his knowledge of the sea-floor maps, navigation, and his own mental calculation powers to safely navigate a large ship out of danger. His fortitude under pressure and his eventual book, The American Practical Navigator, ensure Nat Bowditch a place in history; after his experiences on the ocean, he became a legend in terms of every sailor needing to know his work to operate and survive at sea.