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Perhaps it's worthwhile discussing, in a bit more detail, Frank's discovery 
that short-cut methods are possible, simplifying the clearance of lunars, 
over quite a range of possible geometries. I think he has it right. It's 
highly inventive. But it isn't any sort of replacement for the real thing.

I had asked-

"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?"

Frank's answer was this-

"A whole arc-minute?? Well, quite a large number of cases! I previously 
focused on cases where the resulting error was only a TENTH of a minute, 
and even then there are far more cases than you would expect. It's not just 
small, first-order deviations from the exact vertical circle case that 
work, but much larger deviations.

You asked:
"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?"

There is a very simple test. You compare the difference between the 
observed altitude of the Moon and the adjusted altitude (adjusted to force 
vertical circle geometry). If that difference is smaller than 
6*tan(LD)/cos(h_moon) then any error resulting from treating it as a 
vertical circle case is a tenth of a minute of arc or less. "

Thanks for a direct answer to my question.

So, let's sum up what our navigator has to do, when he decides that the 
geometry is such that Moon and other-body appear to be somewhere near 180º 
apart in azimuth, so a likely candidate for a short-cut, trig-free, lunar.

Having measured sextant altitudes of moon and Sun-or-star, and lunar 
distance between the appropriate limbs-
Correct measured distance for index error and combined semidiameters, to 
get apparent distance d between their centres
Correct altitudes for index error, dip, and semidiameters, to get apparent 
altitudes of centres above the horixontal of Moon m and Sun-or-star s.
From refraction and parallax of other-body, where relevant, applied to s, 
find true altitude S
All this has to be done whether of not a short-cut procedure can be taken.
=================
Next comes the procedure to see if a short-cut can apply-

Find the difference between m and the quantity 180 - s - d, the value it 
must have if the azimuths differed by exactly 180º
Compare that difference with the quantity 6' *tan d / cos m, by calculation 
or by table.
If it's more, the short-cut has to be abandoned, and normal trig clearing 
procedure adopted.
If it's less, the short-cut can apply. In which case-
Discard the calculated value of m, and substitute m' = 180 - s - d.
From refraction and parallax of Moon, applied to m', find true Moon 
altitude M
Then find cleared lunar distance D simply, from D = 180º - S - M.

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

But if the short-cut didn't apply, then preserve the original value of m.
From refraction and parallax of Moon, applied to m, find true Moon altitude 
M
Then find cleared lunar distance D from
cos D = (cos d - sin s sin m) (cos S cos M) / (cos s cos m) + sin S sin M

That final step should present few difficulties to anyone with a modern 
calculator, but was the very devil to calculate using logs, in the days 
when it had to be done that way.

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

I've written it all out to check whether I've understood the procedure. No 
doubt Frank will put me right if I've got it wrong.

It should be clear that the short-cut could save quite a bit of work if the 
appropriate conditions apply. But the test is non-trivial, and if the test 
failed often, aborting that extra work, and necessitating the full 
procedure anyway, then overall there might not be much saving. So it 
depends on the navigator being able to judge well in advance whether an 
observation is a suitable candidate or not. No doubt, that skill could come 
with experience.

But now, consider the situation facing the navigator who has never learned 
the full trig procedure for clearing a lunar. What happens if the test 
fails? He has no alternative but to pack up, and wait for another day. But 
away from the tropics, the opposite-azimuth situation will arise only once 
a month.

That doesn't matter, to a hobbyist, who just wishes to test  his skill when 
the occasion arises. He could go home and try again another day. But Frank 
was imagining himself back in the 1780s, when lunars were starting to be 
used in earnest by professional navigators at sea. They knew how to work 
them. They may not have understood what they were doing, but it was a 
familiar, well-memorised routine. And when they neede a lunar, they needed 
it there and then. Lunars were unavailable for so much of the time anyway: 
when it was cloudy, when the Moon wasn't up, or when it was too near the 
Sun. So such a short-cut procedure wasn't an alternative, to anyone doing 
real navigation. The full trig procedure had to be available, at his 
fingertips, for when it was needed.

Maybe Frank's appearance, with his new procedure, in 1780, would have been 
welcomed by navigators. But I have my doubts.

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.

   

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