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Joel Jacobs asked-

>Within the time constraints that Frank select 20 m, we all agree that you
>don't need >reference to tables to calculate lat. However, it was
>previously stated by George, and >Bill, that it is far more accurate to
>select times of equal altitude that are separated >by an hour or more, to
>which I also agree. But in that case, would you not have to use >tables to
>estimate lat because the azimuth of the body changes so rapidly as it
>>approaches and passes Meridian Passage?

A fair question. Even for lat. only, you would certainly need a table or
almanac showing the declination of the Sun. For long, you would need at
least a table of Equation of Time.

That recommendation of spacing by an hour or more was, of course, for
obtaining long., not lat. Here are some answers to Joel's question.

1. As well as taking altitudes of the sun, an hour or more before andr
after noon  to get an accurate longitude, what's to stop you taking another
observation at LAN (or max altitude; it makes hardly any difference in
deducing lat for a slow vessel) to get latitude at noon?

2. You can use the am and pm altitudes to obtain, and cross, two position
lines, one of which you will have to transfer, to account for your travel
in between. You can also cross a third position-line from a noon sight.
Observations that are well-separated in time from noon give a good angle of
crossing. Of course, to do that you need alt-az tables or some sort of
calculator.

3. In the case of land-travellers Lewis and Clark. who measured and timed
equal am and pm Sun altitudes using a reflecting horizon, from a stationary
standpoint, I have become accustomed to the following calculation-

"Half the sum of those am and pm times then gave the chronometer time at
which the Sun would be at its maximum value, very near to the time of
apparent local noon, except for a small correction which allowed for any
slow change in declination of the Sun. So this provided an accurate value
for chronometer deviation from local apparent time (LAT), and hence from
Local Mean time (knowing the Equation of time). Then, if Greenwich Mean
Time was known, longitude could be calculated from the difference between
LMT and GMT.

Half the difference between those am and pm times gave an accurate value
for the half-interval in time, which can be converted to degrees at 15°
per hour.

Using a modern calculator, rather than logs, to calculate  latitude, a
modern implementation of the procedure given for "Problem 1st" by
Patterson, in Lewis' Astronomical Notebook, is as follows-

where L = latitude, D = declination at local noon, H = hour-angle
(half-interval) in degrees, A = altitude

then L = atn (tan D/cosH) ? acs (sin(atn(tanD/cosH))*sinA/sinD)

the ? symbol indicating that the acs function has two results, equal in
amount but one positive, the other negative.

This expression for L usually has two solutions, one with the Sun passing
South of the observer, the other passing North. It is nearly always obvious
which one is needed."

So two equal altitudes, before and after lunch, provide both lat and long.
However, this was for a stationary observer, in which case there was no
need to account for the distance travelled between the observations. I
think there is no great difficulty in making that allowance, but I am not
familiar enough with that procedure to quote it here.

Of course, log-trig tables are needed to do that job, or a calculator.

4.A procedure for determining lat and long was available right back in the
days of Captain Cook, and was published as part of a fill-in form by Robert
Bishop in 1768, which did the following job.

Given a Sun altitude measurement at noon to provide a latitude, and one
other altitude at either am or pm, with a time-difference between them of
an hour or more that could be measured by a deck-watch (no chronometer
needed). Then using 5-figure logs and what was effectively a haversine
method (though that name was not then known), the local apparent  time
could then be calculated. This would then be used with a lunar (on another
part of the form) to determine GMT, and hence the longitude. Nowadays, a
chronometer would be used in place of the lunar. Again, log-trig tables or
a calculator are needed.

Bishop's form is reproduced (but reduced so as to be hard to read) in Derek
Howse's contribution to "The quest for longitude" (ed William J H Andrewes,
Harvard, 1993). It's a bit more legibly copied into Howse's "Nevil
Maskelyne, the seaman's astronomer", (CUP, 1989), a book I am searching
for. If anyone comes across a copy, anywhere, please let me know.

George.


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