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Peter Fogg defends Bennett's book against my statement that-

"  ...working to a relaxed precision of the nearest 1 minute of arc, rather
than the 0.1 minutes of the normal almanac. Everything, interpolation,
correction, sight reduction, is done to that reduced precision, so these
approximations might sometimes combine to put the resulting position-line 3
miles out, maybe  a bit more"

He responded-

>Having used these tables, and comparing the results with those from a nav
>calculator, I can say that what mostly happens is that the small roundings
>tend to cancel each other out, and the final result is within one minute of
>arc. Occasionally it is within 2 minutes, and only rarely within 3 - when
>this happens I suspect operator error - mine!

Well, his claim, and mine, are perfectly compatible. All I said was that
the errors MIGHT SOMETIMES combine to that effect, and I still hold to
that, though if Peter observes such errors he appears to blame himself and
not the tables. The trouble is that if such errors CAN occur, then for
safety you have to assume that they are indeed present, because you have no
knowledge whether the approximations are, by chance, conspiring together
against you, or are cancelling out (as, to some extent, they USUALLY
will!).

Peter may have noted that I went on to say-

>I think that such a level of precision is entirely appropriate to the
>level of accuracy >with which we can observe altitudes from the unstable
>deck of a small boat, from so near >to the wave-surface.

And I hold to that, too. As long as a navigator is suitably cautious in
applying Bennett's tables, he can get along fine. Trouble will eventually
arise if he relies on his assumption  of a precision which is achieved only
some, and not all, of the time. And that's what Peter Fogg is in danger of
doing.

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

On the serious inaccuracy that I referred to in the azimuth tables, Peter
comments-

>I find these Azimuth Solution tables quick and easy to use.

That's certainly true, but it isn't the point, if they don't give the right
answer.

>The procedure to use when azimuth is within a degree or so of due east or
>west is given and explained.

No, that isn't so. As I said, "Bennett includes instructions for resolving
the ambiguity, but the inaccuracies remain." Methods of resolving the
ambiguity are described in some detail, and these will be needed over a
MUCH wider range of azimuths than Peter's "degree or two". As I said, these
procedures succeed in resolving the ambiguity. But the INACCURACY in the
azimuths near East and West is not discussed, and I think the user should
have been warned about it.

>Then the alternative is given, Weir diagrams, beginning:
>'The accuracy of' (Weir Diagrams) 'is superior to' (Azimuth Solution
>tables).

Yes, that is just what I recommended.

>Remember that these are practical solutions for on-board sailors, somewhere
>else it is noted that an azimuth correct to within a degree or two is quite
>accurate enough for practical plotting purposes.

If it was "an azimuth correct to within a degree or two", nobody would
object. But how would Peter react to azimuth errors of 10 degrees or more?

I can see that it will be necessary to quote an example to convince Peter.

The data that the azimuth table requires is the dec, LHA, and altitude, all
to whole-number degrees. Bennett doesn't say whether the input numbers
should be rounded (to the nearest whole degree) or truncated (by dropping
the minutes), and I have presumed that he intends the former. I will give
two examples, which I admit are intended to show up the problem at its
worst..


Example 1.

dec = 55deg 29', LHA = 54deg 31', alt = 61deg 31'.

These values must then be rounded, to 55, 55, and 62 respectively, before
entering the table..

From the azimuth table we get a value for x of 469, and a resulting azimuth
of 88deg to 90deg (there's no way of telling which).

However, if you make the calculation-

arcsin az = cos dec sin LHA / cos alt

and take the minutes into account, the TRUE azimuth should really be 75deg 21'!

Example 2.

dec = 55deg 31', LHA = 54deg 29', alt = 61deg 29' .

You will notice these values are almost the same as above, but round off
quite differently into whole degrees, which become 56, 54, and 61
respectively.

From the azimuth table, this gives a value for x of 452, and a resulting
azimuth of 69deg.

The TRUE azimuth from the formula above, taking the minutes into account,
is 74deg 51'.

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

You can see that there's only a 30-arcminute difference between the two
true azimuths when calculated out exactly. This is no surprise, because the
numbers that were input are so very similar. But look at the enormous gap
between the two azimuths obtained from George Bennett's azimuth table, 90
deg and 69deg, both a long, long way from the true values. Similar errors
can occur whenever a celestial body is near to East or West.

This is the method Bennett described as "finding the azimuth with an
accuracy of one or two degrees".

These results show that the inaccuracies result from the quantisation to
the nearest degree, combined with the extreme sensitivity of the sin az
formula to small changes in the input values for bodies near East and West.

There's a much better formula which derives az from its tan, but that is no
doubt much harder to implement by a table lookup procedure.

If I have got something wrong, or misrepresented the situation, I'm sure
someone will put me right.

George Huxtable.



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