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    Re: Mendoza's method for clearing lunars.
    From: Henry Halboth
    Date: 2004 Aug 2, 11:14 -0400

    My 1839 edition of Norie's contains a Table XXXV - it is entitled "To
    correct the apparent distance of the Moon from the Sun, a Star & etc. for
    the effects of Paralax and refraction" with entries against Apparent
    Distance across the top and Paralax in Altitude or Distance down the
    side; this Table covers distances to 120-degrees. My 1889 edition of
    Norie's contains a Table XXXV, but only covers distances to 30-degrees. I
    will not open these books as necessary for copying, but would be pleased
    to provide specific entries - if you could copy the work forms indicated
    in your posting, I would also be glad to comment.
    ps My 1902 copy of Norie's Tables alone no longer includes the XXXV
    On Mon, 2 Aug 2004 10:40:04 +0100 George Huxtable
    > 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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