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Ken Muldrew made some interesting observations in his posting a week ago,
commenting on Frank's account about a passage to Bermuda in 1807

Frank had quoted-

| > "As the night approached, the sky became beautifully clear, and shortly
| > after sunset I got my sextant to work. Before the twilight was ended,
and
| > the horizon too faint to admit of the altitudes being taken with
accuracy,
| > I had observed four or five sets of lunars.

and Ken asked-

| What do you suppose he means by "four or five sets of lunars"? The almanac
| would only have 2 stars for a given date. I don't suppose he would
| calculate the distances to other stars without mentioning the fact. Would
| a "set" consist of an altitude, a lunar distance, and the other altitude
| in succession (possibly with only a single observation of each)?

and I wonder whether that was really the case, in 1807. It's true that in
the earliest Nautical Almanacs, only one object to the East of the Moon, and
one to the West, would be offered on most days, just as Ken says. I have an
Almanac for 1864, and by then the Almanac was giving (where available) the
Sun, a planet or two, and 5 or 6 stars, some to East of the Moon, some to
the West, for the navigator to choose between, or make multiple
observations.

I don't know from what date that greater choice was offered. Has anyone
access to an Almanac from around 1807, to see how many lunar distances were
tabulated then?

=====================

Ken went on to say-

| Frank also mentioned, in reply to George, that he agreed that the
| altitudes for the later lunars would not have been calculated without
| noting the fact. This brings up the curious difference between those who
| navigated on land, and those who navigated at sea during this period,
| despite their training (and their textbooks) being essentially the same.
| The land navigators always calculated their altitudes. Even when taking
| lunars using the sun, where the time sight is from a solar altitude, they
| calculate the sun's altitude at the moment of the lunar. They take a time
| sight with every lunar, so there is no question of there being any
| difficulty associated with measuring the altitudes. They often take
| meridian altitudes of stars for latitude, so they seem comfortable enough
| sitting about in the middle of the night looking into their artificial
| horizon. The fur trade navigators were always far enough North to allow
| altitudes with an ordinary sextant. It is something of a mystery.

My comment-

Perhaps it ought not to be a great surprise. After all, mariners usually had
a useful horizon to measure up from, so measured altitudes came rather
easily (though perhaps not so easy, at night). Harder for the inland
traveller, who needed an artificial horizon to do that job. To see stars in
an artificial horizon, a mercury reflector was called for, though a
water-surface would do for Sun or Moon. Depending on the observer's
latitude, it could well happen that at the convenient time for taking a
lunar, one or both of the bodies involved might be above 60 degrees
altitude, so that its doubled, reflected, value was out of range of a
sextant. So the inland traveller had to be able to cope with the situation
where the altitudes couldn't be directly measured, and had to be calculated
from the Almanac.

However, that is in itself quite a complex matter, which I will discuss
after a digression about lunar corrections.

First, let's review why the altitudes of the Moon, and the other-body, are
needed when a lunar distance is measured. Because the makers of the almanac
didn't know where on Earth an observer would be when taking a lunar, they
tabulated the lunar distances as though they had been taken by a fictitious
observer at the centre of a transparent Earth. The lunar distance measured
by a real observer can differ significantly from that tabulated value,
mainly because of the altered direction of the Moon, due to parallax.
Because the Moon is so near, that parallax can give rise to a correction as
great as 1 degree, either way. To a lesser extent, refraction in the
atmosphere changes the observed altitude of the Moon and other-body.  All
these corrections change with the altitude, and can be calculated if the
altitudes of the two bodies are known. They combine in a complex way and can
be precisely allowed for in the operation famously known as "clearing the
lunar distance".

We can estimate that in the worst-case, an error of 1 degree in the altitude
will give rise to an error of no more than an arc-minute or so in the
corrected lunar distance, so that altitudes are not needed to be known to
better than about a quarter-degree or so. It's not a precise measurement
that is called for, then, but it is vital that such a correction is properly
made, because of the high precision that's called for in the lunar distance
itself.

The necessary altitudes can be measured, up from a real or artificial
horizon, or instead they can be calculated, in a similar manner to the way
that you get a calculated altitude, when working a position line. It doesn't
matter at all how those altitudes are obtained, as long as they are accurate
enough.

==========================

A lunar observation tells you nothing about your latitude, only longitude,
so to obtain an overall position some sort of altitude observation, most
easily (if possible) a noon Sun, is called for. So the inland observer has
to be competent to use an artificial horizon, if only with the Sun (the
easiest body). And we will see that he has also has to take some sort of
time-sight, and that again can be an altitude of the Sun, when it's well
away from the meridian. We will assume that he carries some sort of watch,
the error of which is known to within some fraction of a minute each day.
That will allow him to measure time intervals with reasonable accuracy, but
isn't good enough to show accurate elapsed time since his departure. Not a
chronometer, in other words, but at least a useful watch.

We will assume that our observer has had clear enough skies to keep track of
his latitude at each noon, and can interpolate latitude at in-between times
using dead reckoning.

At some time, noted on his watch, our observer took a lunar distance, but
was unable to observe the altitudes of Moon and other-body at the time. He
will have to calculate them, then. How is he to go about it?

The problem is, he doesn't know his longitude (that's what he is trying to
find). Nor does he know his true GMT, or the error of his watch on GMT. If
he did, it would be a simple matter to find longitude. Instead, he has to
derive these things from a lunar distance and a time-sight.

At some time, as close as possible to the moment of taking the lunar (within
a few hours either way), the observer should take a time-sight, noting the
time on his watch. This won't necessarily employ either of the bodies
involved in the lunar distance. It will involve getting out the artificial
horizon, which can be used with a water-surface if it's the Sun being
measured, and taking the altitude of a body when it's well toward the East
or West. From that, knowing its declination and the latitude at that moment,
it's easy to get the Local Hour Angle (LHA) of the Sun or other body being
observed, as has been discussed recently on this list. How did we get the
declination, which changes with time, when we didn't know the GMT? Well, for
the Sun, it changes rather slowly with time, so you could guess, well
enough, at the GMT, to take dec from the Almanac, for a first-try. For
stars, of course, dec doesn't change at all.

Next, we want to get the Greenwich Hour Angle of the body at that moment of
observation. That amounts to the observer's Westerly longitude plus (if the
body is somewhat to his West) or minus (if to his East) the LHA of the body,
in degrees. But we don't know that longitude. No matter; just guess a trial
value that's roughly in the right ballpark. It will all work out in the
wash. That will result in a trial value for the GHA of the body. There's no
reason to think that it's the true GHA; that would be the case only if the
guessed longitude happened to be precise. But that trial GHA will do, for
the present purpose.

Next go to the Almanac (for the right day - be careful!), look up the column
showing GHA for that body, and search for two hourly values that bracket the
trial GHA. By reverse interpolation, find a trial GMT that corresponds to
it. Again, there's no reason for that to be the true GMT for the timesight,
but it will do. Check whether the declination of the body, at that time,
corresponds with the initial dec that was chosen before. If it's
significantly different you may have to change the value for dec and
recalculate.

Now, we have a (trial) GMT for the timesight. From the watch, we know the
time-gap between the moment of the timesight and the moment of the lunar,
which may be before or after. So now, we can get a (trial) GMT for the
moment of the lunar. Now we are on familiar ground. From the Almanac, we can
now get the GHA and dec of the two bodies involved in the lunar.

From the DR, we have the observer's lat at the moment of the lunar. The
trial value to take for his long at that moment is the trial long used for
the timesight, adjusted according to the shift in long in the interval, if
any, taken from the DR. From that lat and long, and dec and RA, we can now
easily deduce the altitude of each body, using tables or computer program or
calculator, using the familiar formula. At long last, we have calculated the
necessary altitudes.

The rest of the calculation then follows as for a traditional lunar. We can
and should now dismiss those trial values for long and GMT, which will not
be used again.

Using those calculated altitudes, clear the measured lunar distance, and
then compare it with the tabulated lunar distance from the Almanac or some
computer ephemeris, interpolating to find out what was the actual true GMT
at the moment of taking the lunar. That then gives the error in the watch.
From that known error, we can now get the true GMT at which the timesight
was taken. The Almanac will provide the true GHA (not the trial value, as
before) of the body used for the timesight. Adding or subtracting the LHA of
that body, as appropriate, provides the longitude.

Looking back on that operation, you can see that calculating the altitudes
has not in any way eliminated the need to measure altitudes. What it has
done is to allow measurement of altitude of a more convenient body at a more
convenient moment, and transferring that information to deduce those
altitudes needed for the correction.

I have gone through all the steps rather long-windedly to show what's
involved, and have little doubt that there are short-cuts to be found by
cleverer readers.

What I have described are the steps involved using a modern Almanac, which
works with GHA. In the heyday of lunars, everything was given in terms of
Right Ascension (RA) and the operations were rather different, and somewhat
simpler.

George.

contact George Huxtable at george@huxtable.u-net.com
or at +44 1865 820222 (from UK, 01865 820222)
or at 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.


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