A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding
From: Frank Reed
Date: 2016 Mar 24, 13:29 -0700
The key equation for a meridian altitude, or Noon Sun, specifically, is
Latitude = Zenith Distance + Declination, or shorter, Lat = ZD + Dec.
Here ZD is the corrected zenith distance, either 90° - altitude, properly corrected in detail for dip, refraction and semi-diameter, or historically, and in many modern circumstance, you can use 89°48' - altitude and count the various corrections as already included.
All three of the angles, Lat, ZD, and Dec, have signs on them, at least if you're a modern navigator. For Lat and Dec, positive is north, negative is south. But ZD also has a sign on it, and I teach it as the "way your shadow points", which I find, and many students have found is memorable as a short rule. For Americans, with our "Groundhog Day", you can tell your students to imagine themselves as the groundhog, and that may plant a visual image in their heads. You look to your shadow to decide what sign goes on ZD. And when you do that, the convention is exactly the same as the others: north is positive, south is negative. So suppose you're in the Caribbean and the date is near the June solstice. Your corrected zenith distance is 10° and your shadow points south. The Sun's declination is 23°. So Lat=(-10)+23 or latitude is 10°, and the result is positive so that's north latitude.
There's nothing new in this, of course. It's a short way of remembering the standard rules. I hate to see navigation students start drawing meridian diagrams, which is just how I was taught this long ago, too, in order to decide whether they need to add or subtract to get their latitude by noon sun. Re-deriving the rule every time is pointless. I should add that this shadow rule fails in a rare, special case: an altitude "below the pole" --midnight sun instead of noon sun. If you're ever in the high arctic, that'll give you something to ponder.