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    Re: Graphs of Lunar Distances.
    From: George Huxtable
    Date: 2010 Oct 1, 15:21 +0100

    There are several curiosities on Frank's responses to Douglas' lunar
    distance observations.
    
    Let's start with this exchange-
    
    Douglas originally wrote-
    "Clearing the distance I used direct calculation with a programmed HP50g
    calculator using:-
    D=arccos(sin S sin M +(cos d - sin s sin m )(cos S cos M )/(cos s cos m))"
    
    To which Frank replied-
    "In case anyone else is wondering, there is really no reason to prefer this
    formula over the usual standard direct spherical triangle solution.
    Unfortunately, this trivial and historically insignificant formula was
    given a status it did not deserve in Cotter's "History of Nautical
    Astronomy" and far too many modern readers have been innocently misled by
    that book."
    
    Douglas, very reasonably, responded- "Well this _is_ the basic spherical
    triangle solution for clearing lunar distance"
    
    And Frank replied- "Well, no, not really --not literally the "basic"
    approach. The "basic" solution of the problem is simple, as follows: apply
    the ordinary spherical trig cosine formula to the triangle consisting of
    the zenith, Sun, and Moon (ZSM triangle), using the observed altitudes and
    the observed lunar distance to solve for the cosine of the angle Z at the
    zenith. This angle Z is the difference in azimuth between the bodies, and
    it is not affected by refraction and parallax in altitude, both of which
    act vertically. Then you take that value for cosZ and with the altitudes
    corrected for refraction and parallax you reverse the process and solve for
    the lunar distance. The result is the cleared distance. The equation that
    you are using can be derived from this algorithm by a sseries of simple
    trig identities and algebraic manipulations, and so can many, many others,
    but there's nothing about it that makes the one you've quoted preferred
    over the "basic" solution as I've just outlined."
    
    ==========
    
    This is most odd. What, exactly, is Frank objecting to? Here's the simple
    basis of the standard formula that Douglas quoted; derived without any
    "trig identities and algebraic manipulations", spelled out in detail as
    follows-
    " apply the ordinary spherical trig cosine formula to the triangle
    consisting of the zenith, Sun, and Moon (ZSM triangle), using the observed
    altitudes and the observed lunar distance to solve for the cosine of the
    angle Z at the zenith", therefore-
    cos Z =  ( cos d - sin s sin m ) / cos s cos m
    "This angle Z is the difference in azimuth between the bodies, and it is
    not affected by refraction and parallax in altitude, both of which act
    vertically", therefore-
    cos Z = ( cos D - sin S sin M ) / cos S cos M
    in which case we can write, directly-
    ( cos D - sin S sin M ) / cos S cos M =  ( cos d - sin s sin m ) / cos s
    cos m
    and it hardly taxes the brain to rearrange that as-
    cos D = sin S sin M +(cos d - sin s sin m )(cos S cos M )/(cos s cos m)
    just what Douglas offered.
    Could anything be more simple, more basic, more direct? It gives the answer
    for D in a single line of calculation. It doesn't calculate Z, or cos Z,
    because it doesn't need to.
    So, whatt is the basis of Frank's criticism of what he calls "this trivial
    and historically insignificant formula"? Where is the "derivation from this
    algorithm by a sseries of simple trig identities and algebraic
    manipulations" that he complains of? It isn't derived from "this algorithm"
    at all. I hope he can find a less-flimsy basis from which to aim his
    criticisms of Cotter.
    Frank wrote- "Furthermore, there is nothing important about it
    historically, despite the fact that it has entered NavList lore (going all
    the way back to 1995) as "Young's formula" probably because it was
    described in Cotter (for no good reason)." Cotter includes a section
    referring to "Young's method", but it does not contain the formula Frank
    objects to. Young's method is indeed quite different. Such is the depth of
    Frank's study of Cotter.
    
    The familiar formula quoted by Douglas is rigorous and (for a spherical
    Earth) exact. The drawback is its inappropriatness for calculations using
    lunars. It was taken as a starting point by many knowledgeable authors,
    including Dunthorne (in his New Method), Borda, Krafft; then bent and
    twisted to exclude the need for additions to be made while the user was
    "into logs".
    
    ===========
    
    Next, let's consider Frank Reed's diatribe against  "A History of Nautical
    Astronomy"., by Charles H Cotter (1968). Invited to suggest something
    better, he proposed a text from 1800 and another in French from 1931. That
    turns out to be quite an endorsement for Cotter: for modern
    English-speakers, his book turns out to be pretty unrivalled, then!
    
    It's certainly true that there are aspects of Cotter's book that are hard
    to defend. In much of the mathematical stuff (and indeed elsewhere) he
    appears to be somewhat out of his depth, particularly in the section on
    lunars. Jan Kalivoda and I, with help from Herbert Prinz, assembled a list
    of known and suspected errors in his text, many years ago, which has been
    added-to over the years, as they have surfaced. No additions to that list
    have been offered by Frank Reed. That list occupies 4 close-typed A4 pages,
    with around 50 known or suspected errors. For anyone who owns a copy, the
    intention is that this list of errors be folded into the flyleaf, which
    should make the book a lot more useful; so it's attached to this mail.
    However, Frank's polemic against Cotter seems not to be directed so much
    against those errors, but at some other aspects that I have failed to
    fathom. Certainly, Cotter fails to get the important distinction right,
    between rigorous and approximate methods of clearing lunars.
    
    In spite of those serious faults, there's nothing around to match Cotter's
    book, and I recommend it as a valuable addition to any listmember's
    bookshelf. If you scan Addall books, you will find several secondhand
    copies around at present, particularly in the US, but none are cheap.
    Expect to pay around $70 or more.
    
    ====================
    
    Frank asked "So why did navigators STOP using lunars after about 1820 on
    British vessels and about 1850 on American vessels?"
    
    What evidence can Frank offer that they did, as early as 1820?
    
    That'll do for now, More later, probably.
    
    George.
    
    contact George Huxtable, at  george{at}hux.me.uk
    or at +44 1865 820222 (from UK, 01865 820222)
    or at 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.
    
    
    
    
    
    

    File: 114072.cotters-errors.doc
       
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