FINDING THE LOCUS OF GEOGRAPHICAL POSITION. 235 
geographical position by a perpendicular distance equal to 
the difference in minutes of arc between the observed and 
deduced zenith distance, and toward the direction of the 
observed celestial body or away from it, according as the 
zenith distance obtained by observational measurement is 
less or greater than the zenith distance deduced by depend¬ 
ence on the estimated geographical position. 
This process of finding the Sumner line has come into 
current use among many navigators under the name of the 
New Navigation, and the present practice is to compute the 
zenith distance and the azimuth of the observed celestial 
body in the following manner: 
A latitude and longitude having been assumed within the 
extreme limits of both as given by the dead reckoning, from 
the G. M. T., as known from the chronometer, the hour-angle 
of the observed celestial body, to an observer situated in this 
assumed geographical position, is now calculated from the 
formulae: 
L. M. T. = G. M. T. ± the assumed longitude expressed in 
time, the upper sign being used when the longitude is east 
and the lower sign when it is west of Greenwich; 
and then when the sun is the observed body: 
Sun’s hour-angle = L. M. T.dt the equation of time ; 
and for any other celestial body: 
Star’s hour-angle = L. M. T. + R. A. Mean Sun — R. A. 
Star. 
With this hour-angle and the two including sides of the 
astronomical triangle, viz., the assumed co-latitude of the 
observer and the known polar distance or co-declination of 
the observed celestial body, the opposite side of the triangle, 
which is the zenith distance or co-altitude of the celestial 
body, is computed from the formulae : 
