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    Impossible lunar example. was: Clearing lunars
    From: George Huxtable
    Date: 2010 Aug 28, 00:43 +0100

    This exchange seems to have diverged into two separate topics, so if nobody
    minds, I propose to split and rename them accordingly. The other will be
    "Short-cut lunars".
    We had been discussing an example of a lunar distance, presented in John
    Hamilton Moore's "Practical Navigaion", which was shown to be geometrically
    impossible in Jane Taylor's "Luni-solar ables". Frank pointed out that this
    example had been presented earlier by Maskelyne, and its defects had been
    publicly aired long before Taylor drew attention to them.
    I then wrote-
    "In investigating Janet Taylor's early works, "Luni-solar Tables", and
    "Navigation Simplified", I have become increasingly disappointed by the
    many flaws to be found therein. I was, at least, allowing the lady a bit of
    credit for her perception in discovering that impossibility in a
    To which Frank replied-
    "George, how generous. You gonna pat the "lady" on the head, too?? :>"
    Wihout such heavy sarcasm, this list would be a rather more pleasant
    Frank then objected that "It's an error to call it an error, without
    Why so? There's a fundamental rule, with spherical triangles as with plane
    ones, that the longest side of a spherical triangle cannot be greater than
    the sum of the other two. By the time it's become equal to that sum, the
    triangle has already degenerated to a line; it can't go further. In
    clearing a lunar, there are two triangles, which share a corner at the
    observer's zenith; the apparent triangle, and the true, corrected,
    triangle. That rule applies to both. In this example, there was a
    discrepancy of 13 degrees, in the apparent triangle. Yes, it's easy to
    envisage a discrepany of the odd minute or two; the result of measurement
    error, but Frank concedes that it's way outside that range.
    Yet he writes-
    "The general issue stands nonetheless (and there were SEVERAL such examples
    in the Tables Requisite): you can have cases where the triangle cannot
    exist --with one side longer than the sum of the other two-- and yet the
    process of clearing the lunar is still valid.
    The first edition of Tables Requisite contains no such examples, that I can
    find. In the Googled third edition of 1802, I can find none either. The
    second edition is not presently available on Google, but Frank has informed
    us of one such example; the one that Moore presumably copied. And now he
    tells us there were SEVERAL such cases. More details, please, Frank.
    And when he tells us that "the process of clearing the lunar is still
    valid", what on Earth can "valid" mean, in such an impossible situation? As
    a mathematical abstraction, that you can work your way through the
    procedure, and arrive at some sort of answer? But what does such an answer
    MEAN, when it corresponds to no measurable quantity? You couldn't draw such
    a geometry on a chart, or on a globe, or see it in the sky. How could you
    check that such a process gave the "right" answer?
    The side-track, about the relative insensitivity of the lunar distance to
    Moon altitude, was irrelevant to the example we were discussing.
    About the correspondence with Maskelyne, Frank wrote- " It's not on Google
    Books, but a small book published by "Nauticus" covering the letters is
    indexed there (no text, just the title). It's something like "Mystical
    Mathematics applied to Moon-hauling". I think he was trying to be funny..."
    I can find a few references on Google (but no scans) to a few books by
    Nauticus or Philo Nauticus of that period, but none with anything like that
    topic in the title. Frank presumes a Maskelyne view as "a transcription
    error of no consequence", but it can hardly be a transcription error, as
    all the triangle quantities must have then been used consistently within
    the calculation.
    Frank points out that if the zenith angle had been calculated as an
    intermediate step, then that would have shown up an impossibility
    straightaway, and indeed it would have brought the procedure to a premature
    end; no bad thing, in this case. But the usual solution works by
    eliminating the zenith angle from the expression, in which case that
    safety-test has been lost. Frank simply tells us to proceed, "just
    calculate cos(Z) and continue", even though the value of cos(Z) is
    significantly greater than 1. And he says "The math works out fine, and the
    results are accurate". Well, in the impossible example we are considering,
    I ask him to justify that statement, that "the results are accurate"? What
    can it mean?
    Frank told us "Whatever the reasons, you'll find more lunars in old
    logbooks taken within 30 degrees of the equator (yes, a bit broader than
    the tropics) than outside those limits." I wonder what those statistics
    were based on?
    The other matter which has emerged, that of special geometries allowing a
    short-cut in lunar calculation, is worth discussing separately, in a new
    thread, which I'll name "Short-cut lunars".
    contact George Huxtable, at  george@hux.me.uk
    or at +44 1865 820222 (from UK, 01865 820222)
    or at 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.

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