Notes on Running Fixes
Under normal conditions, one would expect an error of +/- 0’.3 in the Position Lines. (This error is mainly due to the time recorded under practical conditions.) Land Surveyors achieve accuracy comparable to a GPS using more sophisticated instruments but the same calculations/ method.
Final accuracy is obviously improved by taking more observations. Six star sights will typically provide a fix within 0’.2 of the true position.
Most people adopt some shortcuts in the interest of speed. These have a cost in terms of accuracy. The Sun's Total Correction Tables assume that the Sun's semi-diameter is either 15'.9 or 16'.2. A Sun Sight in April (SD = 16'.0) is immediately in error by 0'.2. Tables are rounded to the nearest 0'.1 which could introduce a cumulative error of 0'05 for every item.
With Star sights, the short interval between the first and last sight means that many people use a single position for all the sights and plot the results without allowing for the vessel's movement. The error is larger than above, but more than acceptable in mid-ocean.
Before Calculators and GPS
The method used until the 1980s was the Haversine Formula with Log Tables. A few navigators, mainly military, used Sight Reduction Tables but most preferred the longer method in the interests of accuracy and flexibility.
The Haversine formula is a rearrangement of the Cosine formula substituting Haversines for the Cosine terms. (Hav(Angle) = ½ x [1 – Cos(Angle) ] ). This makes a calculation using logarithms slightly easier, as the terms are always positive.
Hav(CZD) = Hav(Lat difference Dec) + Hav(LHA) x Cos(Lat) x Cos Dec)