234 
LITTLEHALES. 
whose hour-angle at Greenwich is equal to the longitude of 
A west of the meridian of Greenwich, and B to be any point 
on the surface of the earth whose great-circle distance from 
A is z; then to an observer at B the zenith distance of the 
celestial body in the zenith of A will also be z; and, as B 
may lie in any direction from A, it follows that the zenith 
distance of the celestial body will be the same to an observer 
situated at any point on a small circle of the earth whose 
pole is the sub-celestial place and whose circumference is 
everywhere distant from the sub-celestial place by a great- 
circle arc equal to the zenith distance. Therefore if a navi¬ 
gator, having measured the altitude of a celestial body, 
should obtain its zenith distance, he is certain that at the 
instant of observation his geographical position is some¬ 
where upon the circumference of such a small circle. In 
practical navigation the ship’s position as given by dead¬ 
reckoning always limits the portion of the circumference 
upon which an observer can possibly be to such a small arc 
that, if laid down upon any chart projection, it will not differ 
sensibly from a straight line. Such a straight line is called 
a Sumner line, and, for the same reason that the tangent to 
the circumference of a circle at any point is at right angles 
to the radius drawn to that point, it possesses the property 
of being at right angles to the true bearing of the celestial 
body upon whose zenith distance it depends. 
Reverting to figure 1, let E represent the estimated geo¬ 
graphical position of the observer at the time when the 
altitude is measured which gives the zenith distance z, 
equal to AB. If the observer should compute or by an y 
other means deduce the great-circle distance EA and the 
bearing of A from E (which represent respectively the 
zenith distance and azimuth of the celestial body as they 
would appear to an observer at E), the portion of the small 
circle BB' B" B //r , which represents the actual Summer line 
of position, could at once be laid down on the navigating 
chart, for it would be at right angles to the bearing of the 
observed celestial body and removed from the estimated 
