VOL. LIV.] PHILOSOPHICAL TRANSACTIONS. 153 



lying in the first rule derived from the fluxions of a spherical triangle. To the 

 rules he has here subjoined their demonstrations. 



With respect to tlie usefulness of these rules, he entertains hopes that they 

 will appear more simple and easy than any yet proposed for the same purpose: 

 the last rule, for computing the distance of the moon from a star, though only 

 an approximation, being so very exact, seems particularly adapted for the con- 

 struction of a nautical Ephemeris, containing the distances of the moon from the 

 sun and proper fixed stars, ready calculated for the purpose of finding the longi- 

 tude from observations of the moon at sea; an assistance which, in an age 

 abounding with so many able computers, mariners need not doubt they will be 

 provided with, as soon as they manifest a proper disfxjsition to make use of it. 

 A Rule. To compute tlie contraction of the apparent distance of any two heavenly 

 bodies by refraction ; (he zenith distances oj both, and their distance from each 

 other being given nearly. 



Add together the tangents of half the sum, and half the difference of the zenith 

 distances; their sum, abating 10 from the index, is the tangent of arc the first. 

 To the tangent of arc the first, just found, add the co-tangent of half the dis- 

 tance of the stars; the sum, abating 10 from the index, is the tangent of arc the 

 second. Then add together the tangent of double the first arc, the co-secant 

 of double the second arch, and the constant logarithm of 114" or 2.0569: the 

 sum, abating 20 from the index, is the logarithm of the number of seconds re- 

 quired, by which the distance of the stars is contracted by refraction : which 

 therefore added to the observed distance gives the true distance cleared from the 

 effect of refraction. 



This rule is founded on an hypothesis, that the refraction in altitude is as the 

 tangent of the zenith distance : and the refraction at the altitude of 45 degrees 

 being 5 7*', according to Dr. Bradley's observations, therefore the refraction at 

 any altitude, calling the radius unity, is = 57'' X tangent of the zenith distance. 

 This rule is exact enough for the purpose of the calculation of the longitude 

 from observations of the distance of the moon from stars at sea as low down as 

 the altitude of 10°, for there the error is only 10" from the truth. But if the 

 altitude of the moon or star be less than 10°, the rule may be still made to an- 

 swer sufficiently, by only first correcting the observed zenith distances by sub- 

 tracting from them 3 times the refraction corresponding to them, taken out of 

 any common table of refraction, and making the computation with the zenith 

 distances thus corrected. This correction depends on Dr. Bradley's rule for re- 

 fraction, which he found to answer, in a manner exactly, from the zenith quite 

 down to the horizon, namely that the refraction is = 57" X tangent of the ap- 

 parent zenith distance lessened by 3 times the corresponding refraction taken out 

 of any common table. 



VOL. XII. X 



