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Article from Popular Astronomy (1945) by Paul E. Wylie, "A Simplified
Method of Latitude Determination from Meridian Observations"

"Many texts on spherical astronomy and on navigation recognize four
cases in the solution for latitude of the observation at culmination of
a heavenly body... It is the purpose of this paper to show that the
division of this problem into 'cases' is unnecessary... One simple
formula will be developed, applicable, with simple qualifications, to
all cases of meridian observation for latitude, including both those at
upper and those at lower transit. Once the validity of this formula has
been established, observers may thereafter apply it mechanically, just
as navigators now apply rules in problems of compass conversion."

Wylie defines two angles:

Hx = altitude above the south horizon. E.g., if the body is 10° above
the north horizon, Hx = 170.

Dx = the angle, on the upper branch of the meridian, from the equator
north to the body. Unless the body is below the pole, Dx is declination,
negative if south. If the body is below the pole, Dx = 180 -
declination, negative if below the south pole.

With those angles, north latitude = 90 - Hx + Dx

Examples (upper transit unless otherwise stated):

Altitude 50 in the south, declination north 20. Latitude = 90 - 50 + 20
= +60.

Altitude 50 in the north, declination north 20. Latitude = 90 - (180 -
50) + 20 = -20.

Altitude 10 in the south, LOWER TRANSIT, declination south 20. Latitude
= 90 - 10 - (180 - 20) = -80.

Finally, mirror image of the previous: altitude 10 in the north, LOWER
TRANSIT, declination north 20. Latitude = 90 - (180 - 10) + (180 - 20) =
+80.


http://adsbit.harvard.edu/cgi-bin/nph-iarticle_query?bibcode=1945PA.....53..108W&db_key=AST&page_ind=1&plate_select=NO&data_type=GIF&type=SCREEN_GIF&classic=YES

Clearly, Wylie doesn't have the traditional aversion to signed
arithmetic in navigation. In lower transit observations his formula
seems tedious — the last example requires one complement and two
supplements — but I believe that in practice the great majority of
navigators could ignore the case of lower transit.

My own preference is to take the complement of altitude, which yields
distance from the geographical position of the body. Then apply that to
declination, in the common sense way according to whether you're facing
north or south. That even works for a sight at lower transit, though the
math is more complicated.

   

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