MAY 19, 1923 PROCEEDINGS: PHILOSOPHICAL SOCIETY 215 
PROCEEDINGS OF THE ACADEMY AND AFFILIATED 
SOCIETIES 
PHILOSOPHICAL SOCIETY 
878TH MEETING 
The 878th meeting was held in the Cosmos Club Auditorium on Saturday, 
February 10, 1923, with President White in the chair and 43 persons in 
attendance. 
Mr. G. W. Lirrtenaues addressed the Society on New resources to 
lighten the labor of navigators in finding geographical position from observations 
of celestial bodies. The paper was illustrated by lantern slides and was 
discussed by Messrs. Rup, Lirrrock, and Peters. 
Author’s abstract: Included in the equipment of navigators are instru- 
ments for measuring the altitude of celestial bodies above the horizon and 
also chronometers or watches intended to show, from instant to instant, 
the time of day at the meridian of Greenwich. 
An observer who has determined the altitude of a celestial body at a 
given instant of time has in reality located himself at the end of a radius 
whose length is the zenith distance, or 90° minus the altitude of the celestial 
body, and whose origin or center is the geographical position of the observed 
celestial body, or that place on the surface of the earth which is vertically 
under the observed body at the instant of observation. In the absence of 
knowledge of the precise direction of the radius, the only definite informa- 
tion to be obtained from the observation of the altitude of a celestial body 
at a given instant is that the observer is located on the circumference of a 
circle described by the radius. 
Each separate altitude and corresponding Greenwich time of observation, 
whether it be of a different celestial body or of the same celestial body in a 
different quarter of the heavens, will result in one of these circles of position; 
and it is by the intersection of the circumferences of two or more of these 
circles that the actual geographical position of the observer is fixed. 
Of course, if the observer has migrated in the interval between two ob- 
servations, it will be necessary, in order to find his geographical position at 
the instant of the second sight, to consider the center of the first circle of 
position to be moved in a direction represented by his true course between 
the stations and by an amount equal to the distance between them. 
These circles are called Sumner circles or circles of equal altitude since 
the circumference of each traces out a line on the earth’s surface from every 
point in which the altitude of the observed celestial body is the same at the 
time of observation. In practice it is unnecessary to draw the whole of the 
circumference corresponding to each observation. 
Every part of the Sumner circle is perpendicular to the true bearing of 
the celestial body observed, and therefore the azimuth of the body observed 
is equal to the angle which the Sumner line makes with the parallels of 
latitude on the Mercator chart. Hence, if the latitude and longitude of 
one point in the Sumner line be known, and also the true azimuth of the 
observed position, the line may be drawn on the chart. 
The process by which the longitude corresponding to a given latitude, or 
rather the relative longitude or hour angle, and also the azimuth are calcu- 
lated is simply the solution of the spherical triangle whose sides are the 
estimated co-latitude, the zenith distance or co-altitude, and the polar 
distance or co-declination. These three sides are the data: the results to 
be calculated are the hour angle of the celestial body from the ship’s meridian, . 
and the azimuth. 
