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    Re: Slocum's lunars / Chauvenet
    From: Herbert Prinz
    Date: 2003 Dec 18, 14:31 -0500

    Frank Reed wrote:
    > Slocum probably was using tables keyed to that era's Bowditch. The
    > prefered method then was Chauvenet's which was relatively new at that
    > time. Slocum probably learned a different method when he was a
    > commercial ship captain. The end of the century method may have
    > puzzled him, and I would bet that details of Chauvenet's method would
    > surprise a lot of the mathematically-inclined lunarians who follow
    > this list.
    O.K. I bite. What surprises me is Chauvenet's rigorous concept of
    'approximative' versus the approximate concept of 'rigorous' found in
    other authors. But before I go into the math, I would like to comment on
    the age of the method.
    When it was introduced into Bowditch, more than a decade after Chauvenet
    had died, the method was already a Methusalem. From then, it took only
    half as long for the method to be relegated to the appendix and finally
    to be dropped again, as it took from the first publication in 1854
    (+/-1) to its adoption in "Bowditch".
    The irony is that Slocum was too _young_ to benefit immediately from
    Chauvenet's first publication! However, if Slocum had ever seen any of
    the following at a later stage, he knew the method without having to
    wait for "Bowditch":
    Astronomical Journal, Vol II, (1854?),
    American Ephemeris and Nautical Almanac for 1855 and 1856,
    "New method of correcting lunar distances, and improved method of
    finding the error and rate of a chronometer by equal altitudes",
    Washington, Bureau of Navigation, 1864.
    "A manual of spherical and astronomy, embracing the general problems of
    spherical astronomy, and the theory and use of fixed and portable
    astronomical instruments. With an appendix on the method of least
    squares", Philadelphia, J.B. Lippincott & co.; London, Tr?bner & co.,
    Slocum would have been the right age to see the pamphlet by the Bureau
    of Navigation from 1864 during his training. It is an interesting
    question though, on exactly which manuals the traing in the merchant
    marine was based, how much effort officers made to stay au currant, and
    what pubications they used to this end. (Hopefully not "Bowditch", which
    was always a quarter of a century behind of the newest development.)
    Now for the math. The method is a good counter-example to proof an often
    stated opinion wrong that methods for clearing the distance can be
    divided into rigorous and approximative methods according to whether the
    MZS triangle is solved from the mZs triangle via the common angle Z
    (using the cosine formula twice), or whether MS is derived from ms by
    applying corrections on either end that are estimated by means of small
    plane rectangular triangles with hypotenuses Mm and Ss.
    It is not difficult to trace this wrong opinion back to Cotter, who has
    become an authority on questions of  the history of celestial
    navigation, mainly for the lack of other equally comprehensive
    treatments. To see that this classification is flawed, one only need to
    consider Dunthorne's method. The latter is an approximative method and
    correctly categorized as such by Cotter. Nevertheless it is based on the
    rigorous cosine formula and therefore does not fit Cotter's own
    criterion. What makes the method approximative is the deliberate
    neglection of certain corrections, and omission of terms in simplified
    Chauvenet starts out with a similar approach. He starts with a rigorous
    treatment, even including the elliptical shape of the Earth (!!), as
    well as effects of temperature and pressure on refraction. To my
    knowledge, he is the only one to consider for a sea-method the
    oblateness of the earth, the effect of which can change the final result
    for the distance by typically 6" or 0.1' of arc in mid-latitudes.
    After some substantial kneading, Chauvenet ends up with a seemingly
    awkward representation of the rigorous formula. Only then he
    investigates the significance of individual terms and discards all those
    that can be shown to contribute less than a second of arc or often less.
    What remains thereafter are four terms, A,B,C,D, to be used in two
    corrections to be applied to the apparent distance to yield the true
    geocentric one. The logarithms of these terms fit into a compact set of
    The beauty of all this is that although the method is an approximative
    one from the standpoint of the strict astronomer, it can be rigorously
    shown that the error is below a certain limit that is smaller than the
    accuracy of measurement usually obtainable at sea. This approximative
    method is in fact capable of yielding results that are more accurate
    than those obtained by some procedures based on "rigorous" methods.
    Now I have a practical question (if anything regarding lunars can be
    considered practical). Chauvenet's method relies also on his own table
    of "reduced refraction". This in turn is based on Bessel's refraction
    table. But Bessel's table is no longer used in its original form. The
    N.A. incorporates slightly different tables, and the formula given there
    (and in the Supplement)  is again different. Choosing the right
    refraction table is more of an art than a science (Jan, are you
    listening?); at any rate, it is an empirical process. My question to
    Frank therefore, is:  Have you looked into this aspect? Do you actually
    use the method for practical exercises? It would seem to be a
    contradiction to go all the way correcting for atmospheric conditions
    and at the same time use a table that is officially considered to be
    Coming back to the issue of whether Slocum could have fixed his tables.
    If he indeed used "Bowditch", I would not think so. Having been produced
    by the abstract process I described above, the final tables for log A,
    B, C, D are so arcane that even Chauvenet himself could probably not
    have reverse engineered them, had he forgotten his original procedure of
    devising them.
    Herbert Prinz

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