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)
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