re all the same, then the
observers must all be on the circumference of a circle and _only one_
circle. If they are not on that circle, the altitude they take will be
greater or less than the one in question.

[Illustration]

Now such a circle on the surface of the earth would be very large--so
large that a small arc of its circumference, say 25 or 30 miles, would
be practically a straight line.

Suppose S to be the point over which the sun is vertical and GF part of
the circumference of a circle drawn around the point. Suppose you were
at B and from an altitude of the sun, taken by sextant, you worked out
your position. You would find yourself on a little arc ABC which, for
all purposes in Navigation, is a straight line at right angles to the
true bearing of the sun from the point S. You can readily see this from
the above diagram. Suppose your observer is at H. His line is GHI, which
is again a straight line at right angles to the true bearing of the sun.
He is not certain he is at H. He may be at G or I. He knows, however, he
is somewhere on the line GHI, though where he is on that line he cannot
tell exactly. That line GHI or ABC or DEF is the line of position and
such a line is called a Sumner Line, after Capt. Thomas Sumner, who
explained the theory some 45 years ago. Put in your Note-Book:

Any person taking an altitude of a celestial body must be, for all
practical purposes, on a straight line which is at right angles to the
true bearing of the body observed.

It should be perfectly clear now that if the sun bears due North or
South of the observer, i.e., if the sun is on the observer's meridian,
the resulting line of position _must_ run due East and West. In other
words it is a parallel of latitude. And that explains why a noon
observation is the best of the day for getting your latitude accurately.
Again, if the sun bears due East or West the line of position must bear
due North and South. And that explains why a morning or afternoon
sight--about 8-9 A.M. or 3-4 P.M., if the sun bears either East or West
respectively, is the best time for determining your North and South
line, or longitude.

Now suppose you take an observation at 8 A.M. and you are not sure of
your D.R. latitude. Your 8 A.M. position when the sun was nearly due
East, will give, you an almost accurate North and South line and
longitude. Suppose that from 8 A.M. to noon you sailed NE 60 miles.
Suppose at noon you get another observation.