NavList:

Alex,

It's an interesting issue. I begin my "Modern Celestial" workshops by describing a few things that "we" have today that prior generations did not have or did not use to advantage. Of course there's basic technology like nearly indestructible solar-powered calculators (never mind pocket supercomputers!), but we also have the advantage of several generations of education that are now guaranteed to include basic algebra, and for most people the principal benefit here is skill and confidence in negative numbers. Short answer to your question: when did they start "using arithmetic with negative numbers in celestial navigation textbooks"... only in recent decades. Of course there are rare exceptions, but they had trivial success in the "marketplace" of navigation.

Celestial navigation textbooks and resources up to the very end of positive development (around 1978, which also happens to be the year the first GPS satellite was launched) continued to behave as if negative numbers either didn't exist or were too complicated for average navigators. And in earlier decades, that was largely true. Many students of navigation fifty years ago did not study algebra in their usual "middle school" education. You'll find many simple mathematical procedures mangled by instructions that act as if negative numbers are imaginary numbers [so to speak ;-) ]. Even the simple case of Latitude by Noon Sun which can be summarized by one equation (*), Lat = ZD+Dec, was routinely translated into a confusing recipe of cases. Even today there are navigators and navigation instructors who refuse to deal in negative numbers. Celestial navigation is easy, but the traditional practitioners guarding the castle maintain a system of crazy jargon and needless complexity. They like it that way. :)

Frank Reed

* Well, one equation with a cheat: an extra observational rule to staple a sign on the ZD: if your shadow points south, then make the ZD negative. And there's a bit of hidden detail in both the ZD and the Dec. The ZD, of course, is 90° minus the corrected (!) altitude, and the Declination has to be calculated or interpolated for the time of the sight (nearest few minutes for a Noon sight), but that's standard setup, common to all work in celestial navigation.