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    Re: The Complete On-Board Celestial Navigator Second Edition.
    From: George Huxtable
    Date: 2003 May 3, 22:08 +0100

    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
    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
    On the serious inaccuracy that I referred to in the azimuth tables, Peter
    >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
    >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
    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
    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
    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.
    contact George Huxtable by email at george---.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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