Longhand Sight Reduction

An obscure sight reduction formula derived by a Frenchman named Doniol is the only reduction method that can be quickly done longhand without logarithms. Needed are four place cosine, sine, and haversine tables which can be found in the 1958, 1962, 1966, 1975 vol 2, and 1981 vol 2 Bowditch editions.


http://fer3.com/arc/m2.aspx/Haversine-how-derive-it-Bernecky-jun-2015-g31663


Doniol Formula

Sin h = n - ( n+m )a

Same name:
n = Cos ( Lat-Dec )
m = Cos ( Lat+Dec )
a = Haversine LHA

Contrary name:
n = Cos ( Lat+Dec )
m = Cos ( Lat-Dec )
a = Haversine LHA

rule:
Lat + Dec > 90 use (-)Cos of Complement

When haversine tables are not available then substitute
a = ( 1-Cos LHA )/2