NavList:

Robin,

Your note has been very helpful to me, although the supposed reason for the shift was not quite correct. But I was inspired by your comment to read the original description of Dunthorne method in the 1766 issue of  "Tables Requisite to be Used with The Nautical Ephemeris for Finding the Latitude and Longitude at Sea" again (and much more carefully). An example which takes only the logarithmic difference of the moon in consideration is followed by the remark (page 66):

"Note: If only the first five Figures of the Sines and Logarithms be used, they will commonly determine the Moon’s true Distance from a Star, within 5", or at most 10"; in which Case, the last Figures of the logarithmic Differences [corresponding to the moon’s apparent altitude] is to be omitted, and if the Star’s Altitude be above 5°, the remaining Figures will need no Correction; but if greater Exactness be desired, so that six Figures of the Sines and Logarithms be taken, all the Figures in the Table of logarithmic Differences are to be made use of; and if the Star’s Altitude does not exceed 25°, are to be increased, as in the following Table."

The mentioned Table gives values up to 25° apparent star altitude, because the true logarithmic difference log(cos H / cos h) regarding only refraction is more or less constant between 20° and 90° considering six Figures in the logarithm (0.000122). Thus by shifting this constant part of the star’s logarithmic difference to the moon’s logarithmic difference, the use of the second table is avoided in a wide range of lunar distance observations. Probably the desire for simplification the procedure of clearing the distance was the reason for the same shift in “Stark Tables”.  Unfortunately this simplification works only in the case of a star as second heavenly body in the lunar distance. But this was the only topic of the first description of Dunthorne method. Although the explanation was titled “A new method of computing the effect of Refraction and Parallax upon the Moon’s Distance from the Sun or a fixed Star” the sun was never mentioned in text, example or tables. In the case of sun or planets one has to take the (shifted) logarithmic difference into account for every altitude because of the additional effect of parallax.