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The following was taken from the 1977 Edition of Bowditch article 2027, pages 553 & 554 and may be of interest to any one interested in how the table were constructed:

Another special case of finding latitude is available in most of the northern hemisphere, it utilizes the fact that Polaris is less than 1° from the north celestial pole.   Since Polaris  is never far from the north celestial pole, its observed altitude (Ho) , with suitable correction , is the observer's latitude.   When Polaris is on the upper branch of observer's meridian ( LHA = 0° ) the observer's latitude is equal to the observed altitude minus the polar distance ( p ).   When Polaris is on the lower branch of the observer's meridian ( LHA = 180° ) the observer's latitude is equal to the observed altitude plus the polar distance   ( p ).   When the LAH of Polaris is not  0° or 180°  the Polaris correction is approximately defined by the polar distance ( p ) times the cosine of LHA. Thus the correction is a function of the LHA of Polaris, and hence also of the LHA of Aries, insofar as the difference between these quantities (the SHA of Polaris) can be considered a constant.   Although this method provides sufficient accuracy, a higher degree of accuracy can be obtained by the use of the Polaris correction tables contained in the Nautical Almanac.

The Nautical Almanac tables are based on the following formula:

     Latitude - Ho = p•cos(h) + (p/2)•sin(p)•sin²(h)•tan(latitude)

                    where    p = polar distance of Polaris = 90° -Declination of Polaris 

                                    h = Local Hour Angle of Polaris = LHA of Aries + SHA of Polaris

The value of a₀, which is a function of LHA Aries only, is the value of both terms of the above formula calculated for the mean values of SHA and Dec. of Polaris, for a mean latitude of 50° , and adjusted by the addition of a constant ( 58.8' ).  The value of a₁ which is a function of LHA of Aries and latitude, is the excess of the value of the second term over its mean value for latitude 50°, increased by a constant         ( 0.6' ) to make it always positive.  The value of a₂ , which is a function of LAH Aries and date, is the correction to the first term for the variation of Polaris from its adopted mean position increased by a constant ( 0.6' ) to make it positive.  The sum of the added constants is 1°, so that: 

        Latitude =  corrected sextant altitude -1° +a₀ +a₁ +a₂

The Nautical Almanac table at the top of each Polaris correction page (274→276) is entered with LHA Aries, and the first correction ( a₀ ) is taken out by single interpolation.  The second and third corrections  ( a₁ and a₂ ) are taken from the double entry tables without interpolation, using the LAH Aries column with the latitude for the second correction ( a₁ ) and with the month for the third correction ( a₂ ).