NavList:

I first became acquainted with the quadratic correction formula for lunar distances when I attended the navigation conference at Mystic Seaport that Frank had organized in 2008 and it was essentially the first thing I learned about celestial navigation. It was fairly clear how to derive up to the quadratic term in an ad hoc fashion but going beyond that in a systematic manner wasn’t obvious, at least to me. I was well aware then and now, that the terms beyond quadratic are not needed in practice but it’s an interesting puzzle and challenge. The problem requires eliminating the azimuth difference, Z, and writing everything in terms of the measured lunar distance, d, the altitudes for the star or Sun and the Moon, h_S and h_M, and the angles at their vertices theta_S and theta_M on the spherical triangle. I made little progress until I came across a little known identity called Cagnoli’s equation in Chauvenet’s Plane and Spherical Trigonometry which is a relation between all six parts of a spherical triangle (3 angles and 3 sides). Strangely although the cosine and analog rules are all well-known and given in standard texts, Cagnoli’s equation, which is a natural extension of them, seems to have been lost to history.

In the unlikely event that anyone is interested here’s my derivation of the next order term that I wrote for myself in 2008 and then forgot about until now. Terms of order dh_Mdh_S² and dh_S³ are not given as they are smaller in magnitude but can be obtained by symmetry. Everything is given in radian measure so conversion factors are needed to get things in minutes of arc.

Robin Stuart