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    Mendoza's method for clearing lunars.
    From: George Huxtable
    Date: 2004 Aug 2, 10:40 +0100

    This is a request for some help, please, because my own knowledge and
    information has run out here.
    
    I am following the navigation practices of a captain in the Greenland
    Whaling, who accasionally used lunar distances to determine his longitude.
    
    I have copies of a pamphlet he carried, dated 1816, which was produced by
    J.W.Norie, which had the usual extravagantly-long title of those times, as-
    "Formulae for finding the longitude, in which a method invented by Mendoza
    Rios is used for clearing the observed distances from the effects of
    refraction and parallax, with rules for working the observations.".
    
    It seems that this consisted of a pad of blank forms for the navigator to
    fill in, preceded by a couple of  pages of explanation about how to do it.
    I have copies of some of these completed forms from various whaling
    voyages, and am attempting to work backwards to discover how the job was
    done, in detail.
    
    I have found no more than a mention of Mendoza's method in Cotter's
    "History of Nautical Astronomy", and no details about that method. From
    Norie's explanation it appears to be an approximate method rather than a
    rigorous one (though that doesn't necessarily detract from its accuracy).
    
    A footnote after the explanation states "N.B. In the above Rules, the
    numbers refer to the Tables in Norie's Epitome or Nautical Tables." I have
    my own copy of Norie's, but this dates from much later, 1900, while its
    bound-in tables date from as late as 1914. Even at such a late date, this
    includes tables for working a lunar distance, but unfortunately not the
    tables required for Mendoza's method, which had by then long been
    superceded. Nor does my copy of Raper's (of 1864) seem to carry any
    equivalent tables. So I'm stuck, rather.
    
    The vital table that's missing from that later Norie's is Table XXXV (=35);
    in my edition the tables go straight from XXXIV (=34) to XXXVI (=36). A
    handwritten note on the Mendoza explanation seems to imply that Mackay's
    table LXXII (=72) corresponds, but I don't have a copy of Mackay. The
    Mendoza-method explanation about using table XXXV states- "Enter Table
    XXXV. with the apparent distance at the top, and the Moon's correction in
    the side column, the corresponding number will be the third correction;  in
    the same column, and opposite the difference in corrections, will be found
    the fourth correction."
    
    Another Norie table required was table XXX (=30), stated to correspond to
    Mackay table IX (=9). This was described as for the "proportional logarithm
    of the Moon's correction". In my more modern Norie's, table XXX has become
    simply a table of the Moon's correction, with no mention of proportional
    logarithms, but as proportional logarithms still remain, as table XXXIV
    (=34), by combining two lookups one can get the required answer. So that's
    a problem that can be bypassed.
    
    It's likely that the Bodleian Library will have copies of those earlier
    Norie's, but usually they are very stuffy about photocopying pages from
    their older texts.
    
    So here is my request. I'm asking any Nav-L member who may possess (or have
    access to) a copy of Norie's that's old enough to contain Table XXXV if
    they would kindly let me know how many pages it covers (to assess the size
    of the problem). If it's only a page or two, then if anyone is in a
    position to make a scan and send it to me as a fax or off-list attachment,
    I would be most grateful. Alternatively, similar information about Mackay's
    table LXXII (=72) would be equally welcome.
    
    These difficulties exist only in the section of Norie's pamphlet which
    deals with clearing the apparent lunar distance, by Mendoza's method; the
    rest explains itself well.
    
    For those that are interested, here's a transcript of that section from
    Norie's pamphlet, about Mendoza's method-
    
    ==============================
    
    "To find the true Distance.
    
    1. Add together the apparent distance and apparent altitudes, and take half
    their sum; the difference between the half sum and the Sun or Star's
    apparent altitude call the first remainder: and the difference between the
    half sum and the Moon's apparent altitude call the second remainder.
    
    2. Add together the log sine of the apparent distance; the log. co-sine of
    the Moon's apparent altitude: the log.secant of the half sum; the log
    co-secant of the first remainder; the proportional logarithm of the Moon's
    correction (XXX) and the constant logarithm 9.6990: their sum, rejecting
    the tens in the index, will be the proportional logarithm of the first
    correction.
    
    3. Add together, the log. sine of the apparent distance (already found;)
    the log. co-sine of the Sun or Star's apparent altitude; the log secant of
    the half sum (already found;) the log. co-secant of the second remainder;
    the proportional logarithm of the Sun or Star's correction; and the
    constant logarithm 9.6990: their sum, rejecting the tens in the index, will
    be the proportional logarithm of the second correction. [A footnote states-
    (The sun's correction is the difference of the refraction and parallax in
    altitude.  (IV,  VI) The star's correction is the refraction in altitude
    (IV)).]
    
    4. The difference between the first correction and the correction of the
    Moon's altitude call the difference of corrections.
    Enter Table XXXV, with the apparent distance at the top, and the Moon's
    correction in the side column, the corresponding number will be the third
    correction; in the same column, and opposite the difference of corrections,
    will be found the fourth correction.
    
    5. Subtract the sum of the Moon's correction, and the second and fourth
    corrections from the apparent distance; to the remainder add the Sun or
    Star's correction, and the first and third corrections; their sum will be
    the true distance."
    
    ===================
    
    To me, Mendoza's appears to be a remarkably complex and long-winded method,
    even though it avoided the necessity for using long 5-figure or 6-figure
    logarithm tables. It's not surprising that it failed to survive, in
    competition with (say) Thomson's Lunar and Horary Tables.
    
    In analysing these whaling journals, I could, of course, choose to use
    another method for clearing the lunar distance, one for which all the
    required information was readily available. But I would prefer to follow,
    if possible, exactly the same steps that were taken by this navigator.
    
    George.
    
    ================================================================
    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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