reducing Lunar Observations. ~ 375 
the manner of allowing for the refraction and the sun’s parallax 
that the above authors differ ; all of whom, however, with the 
exception of Lynn, have erred in using in the calculation the 
apparent altitude, whereas the apparent altitude corrected for 
refraction should have been used. On the same grounds the 
usual tables exhibiting the moon’s correction by inspection, 
including parallax and refraction, are erroneously computed. 
By approximative methods are not to be understood me- 
thods that admit of less accuracy, but such as approach the 
truth by a series of which the last terms vanish, or, which 
answers the same purpose, by a gradual substitution in the 
calculation of terms found by a former approximation. The 
approximative methods have, particularly to seamen, that ad- 
vantage above the direct ones, that the trigonometrical calcu- 
lation need only be executed to the nearest minute; and as 
the correction of the distance never can exceed that of the 
altitude, and as its sign, as well as amount, can nearly be esti- 
mated from the position which the observed bodies occupy, 
essential errors may easily be avoided; moreover, they offer 
an easy means of reducing the refraction to the points of the 
limbs brought into contact when observing their distance, as 
shall be shown in what follows: To deduce from the ob- 
served altitude of the upper or lower limb the apparent alti- 
tude of the centre, the horizontal semidiameter must be di- 
minished for refraction by a correction to be taken out from 
the table at the end of this paper: in order to find the vertical 
diameter, the moon’s semidiameter requires moreover the 
usual augmentation on account of parallax. From another 
table, too extensive for our limits here, the reduction from the 
apparent to the true zenith must be taken, which may also 
be computed after the following formula : 
sin ¢ sin h —sindS 2sin¢ 
ss cos2¢ = tang. reduction 
cos h gr AFF ? 6 ‘ 
a*—b* 
neglecting higher powers of a?—d*. This reduction assum~- 
, rT rea aed rt , sing 
ing the ratio of axes nae is very nearly = 1360". i 
(sin g sinh — sin’), where # denotes the true altitude, ¢ the 
latitude, ® the declination, attention being had to its sign. 
Tables for the correction of the horizontal parallax on ac- 
count of the spheroidical figure of the earth are contained in 
several nautical works, or supplied by a simple calculation. 
Let us designate by, 
_ D, the apparent distance of centres; 
d, the distance of the points of contact or observed distance 
of limbs ; 
