The best source of navigation methods is The American Practical
Navigator, originally written by Nathaniel Bowditch in 1802. The
book is commonly called Bowditch, and has been revised and reprinted
by the United States government for about one hundred and fifty
years. I found this method in the 1938 edition on page 106, but
there was no title given. If you know what to call it let me know.

This is probably the shortest method for either a logarithmic
calculation or a calculator, but check the notes below for
some problems.

Bowditch uses the intermediate value called by the Greek letter
Phe, but I will use the letter Q.

This method is similar to the Ageton method (page C2) where a perpendicular
is dropped from the corner of the triangle to the meridian of the
observer (point 1, or point of departure).

When sailing from point one (L1, LO1), to point two (L2, LO2),
Q = distance from pole to the perpendicular, measured along the
meridian of point one.
t = DLO = difference in longitude (or local hour angle)
C1 = course angle at point one
D = distance from point one to point two (degrees)
tan Q = (cos t) / tan L2
tan C1 = (tan t) * (sin Q) / cos (L1+Q)
ctn D = (cos C1) * tan (L1+Q)
To convert D from degrees to nautical miles
multiply D by 60.
To avoid a calculator crash observe the following;
The latitude of the destination (L2) can not equal zero.
(L1+Q) can not equal 90 degrees.
C1 can not equal 90 degrees.
t can not equal 90 degrees.

The solutions to the spherical triangle can be used for calculating
a geographic position from astronomical measurements by substituting
declination and local hour angle for the latitude of point
two and the difference in longitude.

This method was used by the Japanese Hydrographic Office
in H.O. Pub No. 602 (1942)