When you have a calculator or computer handy, sight reduction tables are obsolete. However, if you don't have electronics handy, or don't trust them, or just want to do it the traditional way, then you need the sight reduction tables.
Sight reduction tables are simply pre-computed solutions to various problems in spherical trigonometry.
Each entry in the sight reduction table uses an observer's latitude, a celestial declination, and a local hour angle as inputs, and produces computed altitude (Hc), computed azimuth (Z) and correction factor (d).
To produce tables for all reasonable values of these three inputs would require a library full of books full of tables. Therefore, the inputs have all been rounded to the nearest degree. In this way, the tables are reduced to six volumes. The standard sight reduction set is a publication known as Ho 229. See the National Geospatial-Intelligence Agency for a free download of Ho 229.
(Other formats are also available. The military sight reduction tables for air navigation are available for free download.)
One sample page from the sight reduction table is (approximately) reproduced here. To use these tables, you find the page corresponding to the latitude of your assumed position. Next, you use the declination of the celestial body to select a column. Finally, you find the local hour angle of the celestial body you're sighting on and use that to select a row from the table.
This return three values:
See introduction for sources of published sight reduction tables.
Notes: (below)
This is an editable table. Simply enter the latitude, first declination, and specify same or contrary declination name from latitude.
When I first wrote these pages, the sight reduction tables were a three-volume set. Each page was for a specific latitude, and contained columns for several declinations, and rows for all possible local hour angles. Today, the sight reduction tables which can be downloaded from nga.mil are a six-volume set where each page is for a different local hour angle, the columns are for different latitudes, and the rows are for declinations. Also, values are given to a tenth of a minute precision.
I don't know if the format has changed over the years, if I was originally working from some publication other than Ho 229, or if my memory is just faulty. In any event, this web page now lets you choose between the old format and the new Ho 229 format.
In my opinion, the old format was better, because you didn't have to change page when doing sight reductions for multiple sights.
For example: If the altitude for one declination is 61°25.5219' and the next declination is 61°42.3601', the difference is 16.8381', which these tables display as 16.8'. However, Ho 229 first rounds the altitudes to 61°25.5 and 61°42.4, and then subtracts, leaving 16.9'.
Where this table and Ho 229 differ, this table is correct.
In practice, the error is always less than .05' which probably more accurate than you can hold a sextant in the best of times.