Typically, a series of sights are taken around noon. The navigator can choose to use the highest measured altitude directly, or can plot all of the observations onto a graph and rough a curve through them. The top of the curve can then be used as the culmination altitude.
Included in the Nautical Almanac (and also in some sight reduction tables) is a table for correcting the offset of Polaris from the elevated pole. The navigator simply calculates LHA Aries and enters the table to find the needed correction for the time at which the sight was taken. An accurate latitude can be determined fairly easily this way. A corrected azimuth can also be obtained for checking a compass.
It's practically impossible to accurately shoot the center of the Sun or Moon with a sextant. That's why a correction for semi-diameter is included in the calculations. The navigator shoots the lower limb (typically, but sometimes the upper limb for the Moon) and corrects the altitude for half of the diameter to find the altitude of the center. The SD of the Sun is listed at the bottom of each daily page, but the correction is included in the altitude correction tables for both Sun and Moon.
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