VOL. LII.] PHILOSOPHICAL TRANSACTIONS. 63g 



sent time^ he always allowed for its aberration in longitude, which will sometimes 

 amount to '20", without considering the aberration in latitude, which can be of no 

 consequence in a zodiacal star, such as those are which are always to be used in 

 these observations. The distance of the star from the moon he computed from 

 their longitudes and latitudes, by the two following rules : f^«^i,.,, ^ 



Rule 1 . Add together the logarithmic cosine of the difference of the computed 

 longitudes of the moon and star, and logarithmic cosine of the difference of 

 their latitudes, if they are of the same denomination ; or sum if they are of dif- 

 ferent denominations ; the sum abating 10 frorn the index is the cosine of the 

 approximate distance. — This gives the absolute distance of the moon from the 

 sun, without any further calculation. But in case of a star, it is necessary to 

 apply another rule also. Seven places of logarithms, besides the index, must be 

 used in computing from this rule, and the calculation must be carried to seconds. 

 Rule 2. To the constant logarithm 3.5363, add the sines of the moon and 

 star's latitudes, the v^ersed-sine of the difference of longitude, and the co-secant of 

 the approximate distance just found ; the sum abating 40 from the index is the 

 logarithm of a number of minutes, to be subtracted from the approximate distance, 

 to find the true distance, if the latitudes of the moon and star are of the same 

 denomination ; but to be added if they are of contrary denominations. The 2d 

 of these two rules, though only an approximation, is so exact, that if the latitude 

 of the moon was 5°, and that of the star 1 5^, the error resulting would be only 

 10''' in the distance. Four places of figures will be sufficient in computing from 

 this rule. 



If the distance of the moon from the star thus computed, at the assumed time 

 under a known meridian, suppose Greenwich, agrees with the distance observed, 

 corrected for refraction and parallax, the time at Greenwich was assumed right ; 

 and the difference between this time and the time of observation under the un- 

 known meridian, is the difference of longitude in time between the said meridian 

 and Greenwich ; which is turned into degrees and minutes of the equator, by 

 allowing 15" for every hour, and l" for every 4 minutes of time. 



But if the distance computed differs from the distance inferred from the obser- 

 vation, it must be found by proportion from the moon's horary motion to or from 

 the star, how long time she will take to run over that difference ; whence the 

 time will be found at Greenwich, when the true distance of the moon from the 

 star was the same with that resulting from the observation ; which, compared 

 with the time of the observation by the meridian of the ship, gives the difference 

 of longitude from Greenwich as before. If the distance of the moon from the 

 star computed, agrees with that resulting from observation within lO' or 12', 

 and the distance of the moon from the star be not less than 20° or SO'', the horary 

 motion of the moon in the ecliptic may be taken for the horary motion of the 

 moon to or from the star ; but otherwise the moon's longitude and latitude must 



