A Nautical Almanac is published annually, giving the earth's latitude and longitude of major celestial bodies for each hour of each day.
This information is listed as Declination, which is the degrees above the equator,
and GHA (Greenwich Hour Angle), which is the degrees west of Greenwich, England.
The incremental corrections for the time's minutes and seconds are in the “Increments and Corrections” pages.
Latitude can be calculated using a sextant to measure the angle of the sun from the ocean's horizon at its highest point in the day (local noon),
and then using that angle along with the published current Declination of the sun.
Longitude can be calculated at local noon, if the exact Greenwich time (where longitude is zero) is known. There are 360 degrees of longitude around the world and there are 24 hours in a day, so 360/24 is 15 degrees for every hour difference from Greenwich time. For example, at 15 degrees west of Greenwich, with the sun at local noon, a clock set to Greenwich time would show one o'clock.
Since a noon sighting of the sun with a sextant may not be possible, in the 1870's Marcq Saint-Hilaire published a way to use the data in the Nautical Almanac to calculate the latitude and longitude anytime a navigator can see a celestial body and has the correct Greenwich time.
His "Intercept" method is based on 3 relatively obscure facts:
1) The angle between a celestial body and the horizon would be the same anywhere on a circle around the earth's latitude and longitude of that celestial body.
2) Besides using a sextant, the angle to a celestial body can be found using a method called Sight Reduction.
Sight Reduction can be used for any known latitude and longitude along with the Nautical Almanac's information on celestial bodies.
The method uses spherical trigonometry, but a look-up table or software (such as Sight Reduction Calculator at jdmcox.com) could provide the calculation. The azimuth (true compass bearing) to the celestial body is also found using this method.
3) The difference between two measured angles (in arc-minutes) of the same celestial body is also the difference in nautical miles between the circles on which the two measured angles were taken. These circles represent distances around a location on earth directly under a celestial body.
To use the "Intercept" method to obtain a good latitude and longitude at any time for the current position:
1) Use a sextant to find the angle between the horizon and a celestial body and note the time;
2) find that celestial body's geographic position on earth at that time in the Nautical Almanac;
3) select any position at any latitude and longitude near the estimated current position, and using Sight Reduction,
get the vertical angle from the horizon to the celestial body at that position, as well as its azimuth (true compass bearing).
4) Convert the difference in angles between the sextant sighting and the simulated sighting from Sight Reduction to nautical miles, and
add or subtract that difference on a chart from the simulated position to a new position along the calculated azimuth to the celestial body.
5) This will give a good "line of position" on which you are located, which would be perpendicular to that azimuth, but remember
this "line of position" is simply a segment of a circle relative to the celestial body's earth position.
6) The "Intercept" method must be done again with a different latitude and longitude near the estimated position to get a different "line of position", and then
7) the crossing of the two "lines of position" will provide a good latitude and longitude for your position at the time of your sextant sighting.