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Arthur Pearson described the olden style navigator's routine from times before
the chronometer. After a detailed recapitulation, Bruce Stark concludes:

> You've found latitude and longitude, and the lack of accurate GMT was no
> hindrance whatever in working the observations.

and further down

>
> Modern navigators find this hard to swallow. In the system they've been
> taught, everything is founded on, and must begin with, accurate GMT. They've
> come to accept, as a bedrock truth, that to work observations successfully
> you have to have accurate GMT. If you don't have it the only hope, in their
> view, is to FLOUNDER [my emphasis] toward it by iteration.

If you are able to observe the altitude of the Moon at the time of taking the
distance, you do not need GMT. If, however, the altitude of the Moon at the time
of observation must be computed, one must know the celestial position of the
Moon, and consequently the time at the place on which the almanac is based.
Whether this is expressed in terms of ET, UT, GAT, GAST, LAT in Paris, LAST in
Paris, ... does not matter. If you don't know how late it is at a particular
longitude, one way or another, i.e. if you don't know the phase angle between the
Earth and the sky, no magic is going to help you finding RA, hence LHA, and Dec
of the Moon, both of which which you need to compute its altitude. Local time at
unknown longitude is insufficient.

It is true that the altitude of the Moon is not particularly critical as it only
contributes a second order effect to the final result (a correction to a
correction, as it were). So, it is conceivable that a well chosen estimate of LHA
and Dec of Moon at the moment of  observation (i.e. an implicit guess at time in
Greenwich, as well as your longitude) immediately leads to a sufficiently close
approximation of actual UT, obviating further iteration. The only way to find out
whether the guess was appropriate is to check whether Ra and Dec of the Moon for
the newly found improved UT are reasonably close to the values that have been
used. Only then will the improved UT be close to actual UT. The deliberate choice
to stop iteration after the first round does NOT make the method inherently
non-iterative.

The fact of the matter is that a lunar distance reduction can be worked as a
direct method when observed altitudes of Moon (and/or Sun) are available, whereas
when a computed altitude of the Moon or Sun is involved, there is no such option.
Then, it must always be handled as an indirect method, hence in an iterative way.
There is simply no way to solve the system of non linear equations representing
the lunar distance problem with unknown phase angle between Earth and sky in any
direct way.

In a previous post named "It works - within limits", I pointed to a further
complication of lunar distances involving computed altitudes: They are to be
qualified as running fixes. Like any other running fix, they suffer from bad dead
reckoning between the individual observations. In this particular case, it
concerns the estimated change in latitude and longitude of the observer between
the moments where apparent time is established and the lunar is taken. This is an
entirely different problem; I mention it here only to make it perfectly clear
that this shortcome cannot be alleviated by means of iteration. Iteration helps
bridging a gap when input parameters are only given implicitly. It does not help
when input data is faulty.

Bruce Stark's wording makes it seem as if he considers iteration and indirect
methods a newfangled invention of modern navigators, if not an unsound method
altogether. This would give the wrong picture. To be sure, the sight reduction
method most widely practised in our time, namely the intercept method, IS
indirect and iterative. But is this new? Let's have a closer look at the olden
style navigation. Bruce Stark writes:

> Besides the noon latitude and
> lunar, which took no more work than if you'd had accurate GMT, you've worked
> a time sight twice, using different latitudes. That's exactly what was
> required in order to plot one Sumner line, using an accurate chronometer.

In order to get one Sumner line, one requires one altitude observation plus its
reduction; to get two lines, one needs two altitudes; etc. A noon observation
provides just one other Sumner line and counts for one observation (although it
is actually a series of observations). So, if I count correctly, our 18th century
navigator worked 3 sights where the modern navigator would have needed 2. That's
because the olden style navigator had to work his time sight from the morning
again when latitude became available, as Bruce Stark correctly tells us in his
posting. One may call it "re-working", I call it "iteration".

The idea of iteration is by no means new. Iterative methods are now often
preferred to direct methods for their simplicity. In olden times when
mathematical tools were lacking, they were often the only way to go. For example,
it is preferable to have a method to find apparent time and latitude
simultaneously from any two altitudes. This is more convenient (and safer!) than
a method where a navigator is bound to make an observation at a specific time.
There exists a direct method, called "combined altitude" or "double altitude"
method and the principle was already known to Joh. Regiomontanus. But in
practice, the direct method is by far more labour intensive than the "time sight
- noon sight - iterated time sight"  cycle described above. Therefore, iterative
methods for this more general problem, too, have been proposed as early as in the
18th century: Assume a latitude and find elapsed time between the two altitude
observations. If the time (hour angle) turns out too short, compared with the
chronometer, your latitude was too low, if it's too long, latitude too high. Make
an adjustment and try again. The method must have been in use. Even a practical
man like Lecky offers an improvement to it in "Wrinkles".

Having focused on the task of computing the individual fix, let's zoom out to
look at the ongoing process of navigation as a whole. We find that a passage is,
and always was, one big iteration of taking measurements, making estimates and
basing actions thereon. These actions lead to new situation, new observables. As
new measurements become available, estimates are being corrected and actions
modified. This was even more true in the olden days before modern electronics
took out some (not all!) of the guesswork. Nowadays we call such an iterative
process a feed-back controlled loop. When Nobert Wiener needed a name for an
abstract mathematical discipline to deal with this technique, he chose
"cybernetics" not without a reason: It's the Greek word for "navigation".

Herbert Prinz

   

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