Longitude: How do yo tell what the exact ships time is.
local apparent noon is the point at which the sun is as high in the sky as it is going to get for the day. You find local apparent noon by taking several sites of the sun starting a little before you expect it to occur. Once the sextant redings start getting smaller you know that local apparent noon has occured. you can plot the sextant readings and see the peak...the gmt of the peak tells lets you know how far west of grenwich you are.
An excellent book on this subject is Longitude. It''s a tiny book that can be read very quickly. It tells the story of John Harrison who is the guy who invented the first chronometer accurate enough to allow sailors to determine longitude at sea.
There is another way to determine longitude. Its called the method of lunar distances. The spherical trigonometry is too complicated for the level of education of sea captains in Harrison''s day so everyone figured an accurate clock was the only practical solution to the problem.