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## Finding Longitude At Sea

The problem relates not so much to finding *longitude,* but to finding *position*
at sea!

Much has been written on the development of the marine watch or *chronometer,*
but unless the navigator can find his (and so his ship's) local apparent time
(LAT) __precisely__, the time at Greenwich (GMT) as shown by the corrected
chronometer time is largely meaningless!

Most authors say that apparent noon at sea can be found easily from the
maximum altitude or "culmination" of the sun. That is, at time of
culmination the local apparent time (or LAT) is 12:00 precisely. Comparing
instantly with the GMT (from the corrected chronometer time) yields longitude.
Hey Presto! QED!

Unfortunately, that statement contains one error of principal and several
difficulties!

The error of principal is because the sun's culmination yields a Local
Apparent Time of noon (i.e. LAT = 12:00) but this is not comparable directly
with Local Mean Time:

GMT difference LAT <> Longitude

GMT difference LMT = Longitude

The reason for this is that the sun is not a perfect time-keeper! Throughout
the year the sun crosses the prime meridian at Greenwich, or any other meridian
for that matter, at a slightly different clock time each day. This time
difference waxes and wanes to a maximum and minimum (approximately sixteen
minutes difference) in a well known and predictable way.

This time difference is known as *The Equation Of Time*. Tables of these
values were calculated early in the 17th Century and were available to
clocksmiths and sundial users to enable them to adjust domestic and church
clocks accurately, before good watches were available. The table of values were
of little use to mariners of the period for the same reason - however, once a
good sea watch was aboard our ship, we can now re-write the equation above as:

GMT difference LAT. ± Eq. of T. = Longitude

The above statement is generalised - since GMT is measured as 0 to 24 hours
and longitude is in fact measured as 0-180°W or 0-180°E, there isn't a simple
difference between the two quantities (or even an mathematical one) but it will
suffice for the time being!

The several difficulties referred to above may be broadly described as
operational:

Firstly, it is difficult to measure the precise point of the sun's
culmination! With a low sun - winter noon or ships in higher latitudes - the sun
climbs in altitude slowly, hangs in altitude for several minutes, and then
declines slowly. It is only after *several minutes* of sextant observation
that the navigator decides that he has witnessed the passing of maximum altitude
(i.e. L.A.T. 12:00) and every minute of time represents a ¼° (i.e. 15 minutes)
of longitude error - remember that this error is ± one minute of time.
Therefore the navigator, from this difficulty alone, and assuming that it is
only one minute would not know his longitude with 30 minutes, or in
mid-latitudes within 20 nautical miles or so at least.

Culmination point is further compounded by the vessel's course and speed -
sailing at 10 knots (well within the capability of 18th century craft) -
sail west and the max altitude or culmination will occur actually after 1200
L.A.T. Sailing due East it will be before. Courses with some Northing or
Southing will decrease or increase the altitude erroneously.

Some authors explain away this problem of deciding when local noon occurs by
saying the altitude is taken when the sun bears due south (or north) i.e. on the
observer's meridian! This is sheer deception - marine compasses could not show
bearing to this accuracy and a 2° or 3° error of bearing would be worse than 2
or 3 minutes of time!

If you find this interesting prepare for some spherical geometry and some
Celestial Navigation in the next monograph.

*- 5th January 1999*