Navigation with only addition, subtraction, and tables

In the novel Carry On, Mr. Bowditch, a sailor asked Nathaniel Bowditch to teach him how to do navigational calculations, but the man only knows how to add and subtract by counting on his fingers. He had not heard of multiplication. Bowditch is surprised, but realizes if he made tables of logs of trig functions, then it would be possible for barely numerate sailors to calculate their position. Carry On, Mr. Bowditch is fiction, but it’s essentially factual, and so I imagine something like the conversation above did happen. I thought about how this might work, and it would be difficult. It is true that if someone can add, subtract, and look-up numbers in a table of logarithms, they can effectively multiply. To find the product xy, they would look up the logarithms of x and y, add the results, then use the table in reverse to find what number has a logarithm equal to the sum. Difficulties But there are a couple difficulties in this imagined scheme. First, every calculation would require…