A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding
From: Bill Lionheart
Date: 2020 May 9, 16:18 -0700
The method is given in Todhunter's book Spherical Trigonometry under Art 84 "Two sides and the angle opposite one of them known" (the one known being the azimuth). You can find the book on line from several sources.
There is a fairly complicated description on how to resolve the ambiguity in Art 86. The term "same affection" is defined in Art 65 and it means either both bigger than a right angle or both less than a right angle.
Personally I have only remember seeing this in the spherical trigonometry literature rather than application to navigation/surveying/astronomy, but a quick Google found problems using Napier's analogies in historical civil service examinations on surveying!
Another way to solve the problem is to use the sine rule so you have two sides and both angles opposite to them. Then the cosine rule gives a pair of simultaneous equations for cos of the other angle and side. The result is (19) in Urs Dietrich and Kurt Girstmair Napier’s main application: spherical trigonometry, which is a summary of some of the history and Napier's part in it https://www.collectanea.eu/app/...[...]...pdf?t=1485287133
I assume this is all the sort of thing that is "well known to people who know it", and completely unknown to everyone else! I expect a fair few people on NavList fall in to the first category, and I feel rather I am "giving my grandmother an egg sucking tutorial".