Last night, however, I got one more reminder that there is rarely anything truly original in traditional marine navigation. There have just been too many people working on it for too long. Rarely is something without precedent; it is just rediscovered or reformulated. And so is the case with our handy Small-angle Rule.
Below is a page of an early textbook with the unusual title M. Blundevile, His Exercises, published in 1594. This is in one of eight sections, called the Art of Navigation, Chapter 37. (I am crawling through books like this looking for something very specific, and if I find it, you will be the first to know!)
He is here in a section discussing compasses and related position reckoning and wants (as we do) to offer a way to make estimates of errors. This is the time period when navigators were just learning compass use on a global scale (Columbus had no idea how his compass worked on his famous voyage), and taffrail logs to measure distance traveled through the water were first described just 15 years earlier than this book, in 1580. The reference to Mariner's Card, refers to early form of a Mercator chart.
His angle units here are Rombes, which one learns earlier in the book is what we now call compass points. In fact, it is books like this where we learn where the concept of compass points (11.25º, 1/32 of a circle) comes from. His units are leagues, and since at this time the English were absorbing navigation from the Spanish as they developed their own, he deals with both Spanish leagues (2857 fathoms) and English Leagues (2500 fathoms), but the units do not matter at this point. (England and Spain were more or less at war during this period, so there was much competition in navigation.)
Below is a copy of his angle rule for DR errors. It is not directly our Small-angle Rule, but the principle is the same. That is, we skip the trig, and give specific values that can be scaled. Below this picture is the interpretation of what he means. In short, if you make good one point wrong from what you steered, you will be 19.6 miles off course for every 100 miles you log.
|After running 100 leagues with a course wrong by 1 Rombe, you will be off course by 19 3/5 leagues.|
His Small-angle Rule would then be: Tan (11.25º/2) = [(19 3/5)/2]/100 or
Tan 5.625º = 0.098
which is analogous to our
Tan 6º = 0.1
See how much easier it is to navigate now than it was 419 years ago.