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    Re: Easy Lunars
    From: George Huxtable
    Date: 2004 Apr 29, 21:11 +0100

    Riffling through the dross in my in-box, among the offers for Viagra and
    enlarging my penis (how CAN they tell? I ask), I found this nugget on "Easy
    lunars" from Frank Reed.
    I liked his clear explanations, and his simple approach to estimating a
    land-bound rough-altitude of the Sun. In a flat and gridded city such as
    Chicago, with many shoebox buildings, there will be no shortage of parallel
    horizontal lines to converge at a vanishig-point on the horizon. Perfectly
    good enough in accuracy for estimating Sun altitude, to allow for the small
    correction for Sun refraction.
    Clearly, Frank's proposed method expects the user to have some sort of
    calculator, with trig functions. In that case, I think further
    simplifications might be made.
    Follow his explanation until you get to-
    >And now to clear the distance...
    At this point, we have the observed lunar distance d, which should already
    have been partly-corrected for index error and for semidiameter(s), but not
    yet for the effects of parallax and refraction.
    We have the Moon's altitude m, which should already have been
    partly-corrected for index error, dip, and semidiameter, but not yet for
    the effects of parallax and refraction.
    And for the other-body, which might be the Sun, or a star, or a planet, we
    have its altitude s, again partly-corrected for index error, dip, and (if
    relevant) semidiameter, but not yet for the effects of parallax (if
    relevant) and refraction.
    It's easy to correct m and s for parallax and refraction, by adding an
    amount for parallax (can be up to 1 degree for the Moon) and subtracting an
    amount for refraction. These corrections correspond to Frank's quantities
    dh_Moon and dh_Sun, and he has explained how to obtain them.
    This gives us two corrected quantities, M for the Moon, which because of
    its great parallax will always exceed m, and S for the other-body, which
    will always turn out to be less than s.
    Now, from all this, we need the corrected, or "cleared" lunar distance D,
    which allows for the combined effect, on the lunar distance, of those
    corrections to the altitudes of the Moon and the other-body. And this comes
    straight out from one equation, as follows-
    D = arcCos ((( cosd - sinm sins) cosM cosS / (cosm coss)) + (sinM sinS))
    And that's it! This is the 'cleared' observed lunar distance, to be
    interpolated between the predicted lunar distances which were calculated
    for successive hours, just as Frank explains.
    Of course, to work this on a calculator, all the degrees-and-minutes
    quantities have first to be converted to decimal degrees.
    That expression for D (or its equivalent) can be found in Cotter's "History
    of nautical astronomy" as a stepping stone in deriving other methods;
    Borda's, Young's, and Krafft's.
    I hear you say- "Well, if it's that simple, why didn't they use that
    formula in the 18th/19th centuries?". I will explain. Really, it's because
    they didn't have pocket calculators.
    Just think about the percentage accuracy that a navigator required for
    calculating a lunar. He needed to measure to the ultimate accuracy of which
    his eye, and his sextant, were capable. To better than an arc-minute,
    perhaps to 0.2 arc-minutes after some averaging, in an angle that can
    extend to well over 100deg of arc. So that's to 1 part in about 30,000.
    Even today, there are not many quantities we measure to that accuracy,
    outside a precision workshop. And to avoid any dilution of that accuracy,
    he wanted his calculations to be done to an even better standard, to 1 part
    in 100,000. That's why 5-figure, and even 6-figure log trig tables were
    called for by navigators. In those days using logs was a matter of
    necessity: there was no way a mariner could reliably process such numbers
    by long-multiplication and long-division.
    But there were some terrible snags about using logs, and log-trig tables.
    First, the log of a negative number, or of zero, has no meaning, and trig
    formulae had to be bent and twisted to avoid such numbers arising (which is
    why haversines arose).
    Second, although logs were fine for expressions where trig quantities had
    to be multiplied and divided, if it came to additions or subtractions part
    way through, it was necessary to "come out of logs" by using a lookup
    table, do the additions/subtractions, then take logs again. The extra time
    and trouble, and the extra likelihood of error, that this entailed, meant
    that it was avoided at all costs by further twisting of the trig formulae
    Third, as more accuracy was required of log trig tables, they became bulky
    books, with hundreds of pages to thumb through in each lookup procedure.
    In general, pocket trig-calculators avoid all those problems, and allow us
    to go back to the simpler, basic, underlying trig formulae. Most such
    calculators work trig quantities to quite sufficient accuracy. My own
    ageing Casio works to 1 in 10 billion. If James Cook had been offered such
    a device, he would have welcomed it: though no doubt he would have wanted
    to make yet another circumnavigation, to test it out against his log trig
    tables, before really trusting it.
    Frank goes on to say-
    >The method for clearing a lunar that I've described above is very similar to
    >the old methods of Witchell and Thompson and others which were popular in the
    >19th century (compare Witchell's method in the online copy of Norie --URL
    >below). Those methods depended on logarithms for their calculation of A
    >and B and
    >they used various trigonometric identities to put those calculations in a form
    >suitable for logarithms, so you'll see tangents and cotangents,e.g., instead
    >of sines and cosines, but that's just a re-arrangement of terms. There is no
    >fundamental difference.
    I don't know anything about Witchell's method, but I do have a copy of
    "Lunar and Horary tables", by David Thomson (not Thompson). This must have
    been remarkably popular in its day, as my 1857 copy is already in its 52nd
    edition! This method adds three correction terms to the observed distance
    to clear it, just as Frank's method does. But his method appears to differ
    more from Frank's than he implies. From my limited understanding (it's all
    rather obscure) it appears that the first two of Thomson's terms relate
    only to the corrections due to Moon parallax, and that the refractions and
    any Sun or planet parallax are lumped together into the third correction
    term (which Jan Kalivoda would describe as the "magic" term).
    Thomson's procedure, and Frank's too, offer a clever way of avoiding the
    requirement of extreme precision in the calculation of lunars. They are
    both approximation techniques which take the observed lunar distance and
    simply add some small corrections to it. As those corrections never amount
    to more than 1 degree at most, evaluating those corrections to 1 part in
    1,000 is more than sufficient to get a good enough answer. As a result,
    Thomson's tables became quite thin ones, and easy to handle.  In those
    days, it was a very practical way to correct lunars.
    But I suggest the relative convenience of that method, and its like, have
    now been eclipsed by the pocket calculator, because it can offer all the
    necessary precision at the touch of a button, can handle positive and
    negative trig quantities, and is unworried about mixing in sum terms with
    product terms. For those that have such a trig-function (and inverse)
    calculator, especially one that can be programmed-up with the expression
    given above, the "clearing" of a lunar loses all its terrors.
    Thanks to Frank Reed for a really stimulating posting.
    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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