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Thanks to Frank for providing background information about the "impossible"
lunar problem stated by Moore in his "Practical Navigator", and tracing its
source back to Maslelyne's 1780 edition of his "Tables Requisite". All that
was quite new to me. I can add that it didn't appear in the first edition
of "Tables Requisite", of 1767.

In investigating Janet Taylor's early works, "Luni-solar Tables", and
"Navigation Simplified", I have become increasingly disappointed by the
many flaws to be found therein. I was, at least, allowing the lady a bit of
credit for her perception in discovering that impossibility in a
set-problem. But it seems that even that should be denied her, as the error
had been publicly aired, half a century earlier.

Frank wrote- "There was a vitriolic exchange of letters to the editor of
the London Gazetteer back in the 1780s between an anonymous complainer who
signed himself Nauticus and none other than Nevil Maskelyne. You see, it
was Maskelyne, the Royal Astronomer and great lunarian himself, who
originally published this inconsistent triangle in the Tables Requisite
back in 1780. And he answered the complaints in some detail and with
considerable anger, too. I've got a copy of this exchange of letters
somewhere, but I'll have to dig it up."

I hope Frank's digging is successful, or he can provide a reference, as
that sounds rather interesting. I wonder what was the actual date of that
exchange, "back in the 1780's", as Moore seems somewhat lax in reprinting
the same problem as late as 1795.

But I find Frank's words a bit puzzling, when he writes- "... But she's
wrong. Well, no, not really wrong... she's just missed the point. The
process of clearing lunars is rather robust mathematically, and
observational error in the altitudes is no problem, up to a point --even if
it seems to yield an impossible spherical triangle (though not as extreme
as in this case). Also, inconsistent data don't really enter into the
process of clearing as outlined here. It doesn't "hurt" the examples,
though it certainly could confuse the student without some kind of comment.
Even a footnote would help."

It seems to me that the point she raises is a valid one, even if she wasn't
the first to raise it. An impossible example is of no value. What does
"rather robust mathematically" mean, in this context? Only that if you work
your way blindly through the procedure, you will get some sort of answer
(as Maskelyne and Moore both did), without meeting along the way a sin or
cos greater than 1, or a square root of -1. But is it the RIGHT answer?
That question becomes meaningless, if it can't be related to a real-life
situation, or set up as a physical or mathematical model.

Frank is right, that navigational authors of the day did their best to
denigrate or pick holes in the work of their competitors. In general, they
were covering very similar ground, and emphasised any point of difference
wherever they could. In Taylor's "Luni-solar tables", for example she
emphasised as a Big Thing that every other author was in error in
neglecting the difference in latitude calculated from a "vertical" line
towards the Earth's centre, compared with that defined by the direction of
gravity, and proposed that all latitudes should be corrected accordingly.
She seems not to have realised that latitudes were consistently defined
according to the direction of gravity, and charts had been made
accordingly. That notion was quietly dropped in her subsequent works.

However, I don't see how those words of Moore, that Frank quotes, would
provide any answer to her criticism of his impossible example.

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

Now we come to quite different ground, a matter which Frank has raised
before. It is quite true that special geometries arise in which longitude
can be derived from a lunar distance observation, without involving trig.
And it's also true that the processing of lunars in that way may be
tolerant of quite wide deviations from those exact geometries. But how
tolerant? How much deviation from that geometry can be accepted, and still
keep within an error-band of an arc-minute or so, outside which a lunar
loses much of its value?

So, what criteria does our navigator adopt, in order to decide whether such
a short-cut is acceptable, or whether a full-blown trig solution is needed?
Unless he is particularly knowledgeable about lunars, he will be unable to
make that decision, except in the rare situations when that exact
special-geometry arises. What are the rules-of-thumb that he can use? Or
would he be better off, applying the traditional trig technique in every
case, whether or not a short-cut might have sufficed? After all, in the
days of lunar distance, trig, logs, and tables were the regular diet of a
navigator, and they had no fear of such matters; unlike Frank's students of
today.

Frank quotes the comment- "... clear lunars, at least in tropical seas when
they were, in fact, most useful," What, I wonder, rendered lunars less
useful in higher latitudes? That reasoning escapes me.

George.

contact George Huxtable, at  george@hux.me.uk
or at +44 1865 820222 (from UK, 01865 820222)
or at 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.
----- Original Message -----
From: "Frank Reed" 
To: 
Sent: Wednesday, August 25, 2010 3:55 AM
Subject: [NavList] Re: Clearing lunars


George, you wrote:
"And indeed, Janet Taylor has got it absolutely correct. Moore has set
himself a quite impossible example. With the altitudes that he has given
for the two bodies, there isn't nearly enough spacing across the sky, in
any possible geometry, to achieve such a large value of lunar distance.
It's a contrived example, that can't possibly happen in real life. Yet he
has gone blindly through the motions of making the corrections to clear
that absurd value, by two different methods, and indeed, got the same
result in each case."

I never noticed Taylor's comments on this. But she's wrong. Well, no, not
really wrong... she's just missed the point. The process of clearing lunars
is rather robust mathematically, and observational error in the altitudes
is no problem, up to a point --even if it seems to yield an impossible
spherical triangle (though not as extreme as in this case). Also,
inconsistent data don't really enter into the process of clearing as
outlined here. It doesn't "hurt" the examples, though it certainly could
confuse the student without some kind of comment. Even a footnote would
help.

Taylor blamed Moore for the inconsistent example, and she added (copying
from your post):
"I here observe, that according to 20th 12 Euclid, it is impossible and
absurd that one leg of a spherical triangle should be greater than the
other two".

Though Moore was long dead by the time she was writing these words, Moore
already had an answer for her. In one my talks at Mystic in June, I spoke a
bit about some of the feuds among authors and publishers of navigation
manuals. They were, after all, vying not merely for public acclaim and
bragging rights, but also for profit. And, of course, celestial navigation
has long attracted pedantry and pomposity for a variety of reasons. We all
have encountered people who know just enough navigation to consider
themselves experts and geniuses when they still have a very long way to go.
Moore knew this, and in the introduction to the "New Practical Navigator"
from 1794 (or earlier?), Moore wrote:
"I am well aware that there are persons, who, to shew there own superior
abilities in an obscure Club, will quibble and carp at some parts, and say,
that they see nothing new, &c. To such Critics it may be answered, that a
Triangle was a Triangle before the days of Euclid, and so it is now..."

So take that, Janet! And it wasn't his example anyway...

George, you wrote:
"Much of the impact of Taylor's point, made in 1833, is however lost,
because that mistake occurred  some 38 years earlier. By the time Moore's
18th edition of 1810 (also on Google) had appeared, that erroneous example
had been dropped."

Not to worry. There was a vitriolic exchange of letters to the editor of
the London Gazetteer back in the 1780s between an anonymous complainer who
signed himself Nauticus and none other than Nevil Maskelyne. You see, it
was Maskelyne, the Royal Astronomer and great lunarian himself, who
originally published this inconsistent triangle in the Tables Requisite
back in 1780. And he answered the complaints in some detail and with
considerable anger, too. I've got a copy of this exchange of letters
somewhere, but I'll have to dig it up. Moore, a decade later, was simply
copying from the T. Reqs. By 1833 when Taylor was writing, it was already
long after the heyday of lunars in Britain (and near the end in America)
and the history was already being lost.

This anonymous Nauticus, by the way, also made the point that one could
very easily clear lunars, at least in tropical seas when they were, in
fact, most useful, by waiting for vertical circle circumstances. When the
Moon and the other body are aligned in a vertical circle, either both in
exactly the same azimuth or both on exactly opposite azimuths, the process
of clearing is extremely simple. You just add (or subtract) the ordinary
altitude corrections --no spherical trigonometry required (and you may
remember a similar approach to clearing star-star distances published over
a century later). What Nauticus didn't realize, quite ironically
considering his complaint, is that the process of clearing lunars is so
accepting of errors in the altitude of the Moon that a great many cases of
clearing lunars, significantly distant from vertical circle conditions, can
be converted to the ideal vertical circle case (which I outlined in a
NavList post two years ago: http://fer3.com/x.aspx/05622). Even if the
original observations are inconsistent, as above, within certain broad
ranges the altitude of the Moon can be dropped and replaced by one
calculated (by simple subtraction) under the assumption that the two bodies
are actually aligned in a vertical circle. Then we do the clearing, and the
results are every bit as good as if the full trigonometric solution had
been worked out. As I joked in Mystic at June, I need a time machine to go
back and fill everyone in on this... :)

-FER


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