الحل:

Step 1: determine your latitude.
There are (at least) two possible techniques.

1. Measure the altitude of Polaris above the northern horizon, using the sextant.
This is approximately equal to your latitude.
(Polaris, the "North Star", lies very close to the North Celestial Pole.)

There are various problems with this.
     Firstly, if you are in the southern hemisphere, Polaris will be below the horizon!
     Secondly, you need to carry out the measurement in nautical twilight,
while it is still light enough to see the horizon,
and Polaris is only a second-magnitude star,
so it may not appear bright enough to measure accurately.
     Thirdly, Polaris does not lie exactly at the North Celestial Pole,
so your result could be nearly 1 degree in error.

2. So, as an alternative,
measure the altitude of the Sun at midday, using the sextant.

Knowing the date, calculate the declination of the Sun
(it varies sinusoidally,
with a period of 1 year starting at the spring equinox,
and an amplitude of 23.4 degrees.)

The midday altitude, when the Sun is on the local meridian,
is composed of:
the height of the celestial equator above the southern horizon (equal to the co-latitude)
plus the height of the Sun above the celestial equator (its declination).
(If you are in the southern hemisphere,
the celestial equator will be closer to the northern horizon;
in this case its distance from the southern horizon, the co-latitude,
will be greater than 90°.)
Knowing the altitude and the solar declination,
calculate the co-latitude and hence the latitude.

If the sextant can be read to an accuracy of a few arc-minutes,
you should correct your reading for refraction.
The apparent zenith angle of an object z' is greater than its true zenith angle z
by the value k tan(z'), where k is approximately 1 arc-minute.

Step 2: Determine your longitude.
Again there are (at least) two possible techniques.

1. During nautical twilight,
if you can locate a star whose celestial coordinates you know,
measure its altitude above the horizon using the sextant,
and note the time (GMT) using the chronometer.

Knowing the star's altitude, its declination, and your latitude (previously determined),
calculate its Hour Angle
by applying the cosine rule to "the" Astronomical Triangle.

Knowing the star's Right Ascension,
calculate the local sidereal time of the observation
(Local Hour Angle = Local Sidereal Time - Right Ascension).

Knowing the date,
calculate the Greenwich Sidereal Time
corresponding to the Greenwich Mean Time of the observation.
GST is equal to GMT at the autumn equinox,
and GST runs faster than GMT by one day in 365.25 days.

The difference between the Local Sidereal Time (from your observation)
and Greenwich Sidereal Time (from the chronometer)
is your longitude east or west of Greenwich.

2. Failing a star with known coordinates, use the Sun.
Note the time (GMT) when it reaches its greatest altitude:
this is midday, Local Apparent Time.

Use the formulae given in Smart's Spherical Astronomy
to calculate the Equation of Time on that date.
(Or derive it from first principles:
allow firstly for the non-uniform motion of the Sun around the ecliptic (Kepler's Second Law);
then allow for the fact that the ecliptic is tilted to the equator.)

Add or subtract the Equation of Time to your Local Apparent Time,
to obtain Local Mean Time.
The difference between Local Mean Time and GMT
is your longitude east or west of Greenwich.

Step 3:
Tear a strip of paper from the title-page of Smart's Spherical Astronomy
to write a message giving your latitude and longitude.
Launch it by carrier-pigeon and wait to be rescued!

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