plorer must fall back on the method of Sumner's lines, and fortunately they can be applied with special facility in the neighborhood of the pole.
Let us suppose then that an explorer is approaching the north pole in the neighborhood of meridian 120 degrees. (See Fig. 2, where the outer circle represents a circle one degree from the pole, and the radiating lines are the meridians, 0 degree being that of Greenwich.) He determines the altitude of the sun when by his chronometer, let us say, it is in longitude 30 degrees. He now works out his latitude on the supposition that he also is in longitude 30 degrees; suppose his results give an apparent altitude of 89 degrees 50 minutes. He lays off that latitude on the 30th meridian at A, and draws a straight line
AA' at right angles to the latter; this line will practically coincide with a part of the circle at all of whose points the sun has the observed altitude at the time the observations were made; his position is therefore somewhere on this straight line, and, guessing about how far he has traveled from his last determined position, he can estimate roughly where he is; but if bad weather has prevented observations for several days, or the unknown drift of the ice has been strong, he might be many miles wrong.
If he should wait for six hours and make another similar observation of the sun's altitude when it is on the 120th meridian, he would determine a second line on which he would be; his true position would then be at the intersection of these two lines. If the second observation determined an apparent latitude of 89 degrees 40 minutes, he would lay off this latitude on the 120th meridian, draw a straight line, BB', at right angles to the latter, and his true position would be at B;
