The calculation is done using the haversines formula.
The pulse shape is usually assumed to be of haversine, half-sine or some other tapered rectangular shape.
For all intents and purposes, the haversine is the most accurate way to calculate distance on a sphere.
Regardless of whether you choose to use the sight reduction tables or prefer the haversine formula you will require a nautical almanac.
By the device of using versines instead of haversines in these tables it will be found that solutions of the astronomical triangle will be correct to within one minute of arc.
Origin
Late 19th century: contraction of half versed sine.