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Vic Fraenclel wrote-

>While at amazon.com, it was suggested that I might like to consider "The
>Complete On-Board Celestial Navigator" by George Bennett. I read the first
>couple of pages. The book seemed interesting. Has anyone actually purchased
>the book? If so, how about a comment? The book is only $19.57 + shipping.
>
>Any enlightenment will be appreciated.

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

George Huxtable responds-

This book has been discussed before on this list, starting 3rd May 03,
thread subject-
"The Complete On-Board Celestial Navigator Second Edition"

I have a copy of this book. It is good in parts, but VERY bad in others. It
presumes that the celestial nav. positioning from a small boat is only
going to be good to a few minutes of arc. Therefore, much space and effort
may be saved by restricting the precision of the 5-year almanac it
contains, and the tabular calculation of intercept, to the nearest
arc-minute; not the 0.1 arc-minute that the Nautical Almanac chooses to
work to.

I have no quarrel with this approach. It seems to me eminently realistic,
in the difficult environment of a small craft. The overall inherent errors
can accumulate to provide an intercept to within a few arc-minutes of the
true value. As long as this limitation is recognised and understood, it
won't give rise to danger.

However, to establish a position line, both an intercept and an azimuth are
required. And in my opinion Bennett's way of obtaining azimuth is badly
flawed.

Bennett provides a simple way to calculate the azimuth using a lookup
table, for which he claims it is "one of the simplest techniques for
finding azimuth with an accuracy of one or two degrees". If it met that
claim, than that would be perfectly acceptable. At certain azimuths,
perhaps it does meet it. But at other azimuths, it can give a result that's
in error, by as much as 15 degrees, enough to throw a position-line wildly
out.

My criticism of this azimuth table caused Peter Fogg to argue-

>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.

And I responded-

"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."

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

George Bennett responded-

"Discussion of the Azimuth Tables in The Complete On-Board Celestial Navigator.

With reference to the examples used by George Huxtable:

(1)   If the LHA is 54?, the azimuth is found from the opposite side of the
table, see explanation p19, Step 2.

(2)   The data in the examples is incomplete. To resolve the azimuth
quadrant ambiguity, the procedure via the Prime Vertical Altitude should be
followed. The tangent formula, heeding the signs of the numerator and the
denominator, does not have  this disadvantage. The Weir diagrams are also
free from this defect.



                                  Tan Az =                 -Sin LHA
.

                                                      Cos Lat*Tan Dec -Sin
Lat*CosLHA



                          Lat + N, -S    :      Dec +N, -S     :
LHA  0? - 360?)



In the two examples chosen to highlight the shortcomings of the Azimuth
Table all three variables are in error by 0.5? (? 1?)and the circumstances
are in the vicinity of the Prime Vertical. In these extreme, but possible,
situations the azimuth derived from its sine is somewhat uncertain as will
be seen from an inspection of the Table. If, however, the Tables are
interpolated (X=460) the azimuth is found to be 255? or 285? (not 075? or
105?) which compares favourably with the results from direct calculation of
255.3? and 254.8?.

The user of the book is not informed that this situation can arise. In the
examples given in the book it is implied that all values are rounded off to
the nearest degree. I have used the tables on innumerable occasions,
checking the results by calculator, without this problem occurring.
Nevertheless, I accept that a note to this effect should be included. I
thank George Huxtable for drawing my attention to this situation."

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

Dack to 3rd Sept 03.

George Bennett correctly drew attention, above, to my misreading of the
tables in deriving a Westerly azimuth where it should have been an Easterly
one (sorry about that!). But that, as Bennett accepts, does not affect the
criticism of inaccuracy.

Bennett points out that my examples are "extreme, but possible situations"
.. ""in the vicinity of the Prime Vertical". Well, they are about 15
degrees away from the prime vertical, so are hardly "extreme" examples. He
points out, correctly, that if interpolation is used then a much better
answer will result, but forgets that the instructions state specifically-
"No interpolation is required, and it is one of the simplest techniques for
finding azimuth with an accuracy of one or two degrees".

If Bennett has indeed included a note with his book to inform users about
this problem, this may help prospective purchasers (such as Vic) but not
present owners. Armed with that warning, they will be in a better position
than others who trust that azimuths will always be within one or two
degrees, and who may find themselves in danger as a result.

There are other ways of finding an approximate azimuth direction, including
the Weir azimuth diagram, which is included in the book.

George Huxtable.

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