VOL. LII.] THILOSOPHICAL TRANSACTIONS. 037 



is practicable at sea, and also that so far from being less simple, it is more so than 

 the other method ; for the additional observations that it requires are very easily 

 made, and even the error of a degree in the altitudes would seldom be of more 

 consequence than an error of a minute, in taking the distance of the star from the 

 moon ; so that an error of 10' or 15' in the altitudes would be of no great pre- 

 judice : but with respect to the facility of the calculations, there is no comparison 

 between the methods, the latter being much less intricate, and much more con- 

 cise. The Abbe de la Caille requires the altitude of that part of the moon's limb 

 from which the distance of the star is taken ; but as at sea we can only take the 

 altitude of the moon's upper or lower limb, an allowance might be made near 

 enough, by estimation of the eye, for the difference of altitude between the moon's 

 upper or lower limb, and that part of the limb from which the distance of the star 

 is taken, he generally added the semidiameter of the moon to, or subtracted it from 

 the observed altitude of the lower or upper limb, in order to have the apparent 

 altitude of the centre, and he found the apparent distance of the star from the 

 moon's centre, by adding or subtracting the moon's horizontal semidiameter, 

 augmented according to her height, to or from the observed distance of the star 

 from the moon's nearest or remotest limb. > 



This method will be exact enough, if the altitude of the moon or star be not less 

 than 5°. Having thus got 3 sides of the spherical triangle formed by the moon, 

 the star, and the zenith ; namely, the apparent zenith distance of the moon, the 

 apparent zenith distance of the star, and the distance of the star from the moon, 

 he finds the effect of refraction and parallax, in altering the apparent distance of 

 the star from the moon, by the two following rules. 



Ru/e 1 . To find the effect of refraction in contracting the apparent distance of 

 two stars, or of the moon and a star. 



Add together the logarithm-tangents of half the sam, and half the difference of 

 the two zenith distances, the sum abating 10 from the index is the tangent of arc 

 the first. To the logarithm-tangent just found, add the logarithm-cotangent of 

 half the distance of the two stars, the sum abating 10 from the index is the tan- 

 gent of arc the 2d. Then add together into one sum the logarithm-tangent of 

 double the first arc, the co-secant of double the 2d arc, and the constant loga- 

 rithm 2.0569 ; the sum abating 20 from the index is the logarithm of the num- 

 ber of seconds required ; by which the distance of the stars, or of the moon and 

 stars, is contracted by refraction : which therefore added to the observed distance, 

 gives the true distance cleared from refraction. 



This rule may be made universal, so as to serve with equal exactness almost 

 down to the horizon, if the apparent zenith distances be diminished by 3 times 

 the refraction belonging to them, found from any common table of refraction, 

 and the computation be made with the zenith distances thus corrected. But if 



