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    Re: Lunar distances - shot clearance methods
    From: George Huxtable
    Date: 2004 Sep 12, 12:19 +0100

    Henry Halboth revisits Arnold's lunar distance method, to tie up some loose
    ends.
    At the foot of this posting, I will restate my earlier transcription of
    Arnold's text into emailese, as quoted by Henry, but amended to correct a
    few things that went wrong with his original text.
    
    I commented on Arnold's simplifications of Mendoza, on 5th Sept., as follows-
    
    "This seems a worthwhile simplification on Mendoza's method..
    
    However, it comes at a price. Arnold has had to supply separate tables for
    correcting Moon, Sun, and stars, though he can avoid the need to use
    separate refraction and parallax correction tables.
    
    And Arnold has lost some flexibility in this simplification. When Mendoza
    calculated refraction, it was possible, if he thought fit, to apply
    corrections for a non-standard atmosphere. If a lunar was taken to Venus or
    Mars, which require non-standard parallax corrections, that parallax, if
    known, could readily be applied. (Does anyone know when lunar distances to
    planets started to appear in the Almanac?)
    
    As I see it, however, Arnold's method, including standard refraction and
    parallax in his tables I, II, and III, was inflexible in that it would have
    been unable to adapt to such requirements."
    
    And Henry has responded-
    
    >As regards Arnold's tables being somewhat restrictive by reason
    >of constraints in altitude correction, please note the use criterion
    >to be the bodies apparent altitude or the correction thereto; how either
    >is calculated is not mandated and may be at the user's option -
    >whatever refinement in the way of temperature and latitude
    >corrections for refraction and parallax may certainly be applied
    >at discretion.
    
    ==================
    
    Reply from George-
    
    Well, I don't wish to make a "big thing" of this resulting lack of
    flexibility, because I don't regard it as an important matter in practical
    terms. But I question Henry's argument here.
    
    The difficulty is that the altitude correction of the Moon (and the same
    applies to the Sun or star) enters into Arnold's calculation, not once but
    twice.
    
    Taken from the transcription below is the sentence-
    
    "Add to the apparent distance the first correction and the correction of
    the sun or star's altitude, and subtract the sum of the second correction,
    and the correction of moon's altitude will be the corrected distance."
    
    I think that may perhaps be slightly garbled, but its meaning is clear
    enough. Perhaps it should really have ended something like "...and subtract
    the sum of the second correction and the correction of moon's altitude, and
    the result will be the corrected distance." No matter.
    
    As far as that sentence is concerned, Henry is quite correct. That
    "correction of Moon's altitude" can be worked out by the navigator as he
    chooses, and he may include, or else disregard, temperature / pressure
    corrections to refraction (same for Sun or star).
    
    BUT that altitude correction (which combines refraction and parallax)
    enters also into Table I (and table II or III, as relevant). Arnold states-
    "Enter Table I with the Moon's apparent altitude and horizontal
    parallax, and take out the corresponding logarithm, which place
    in the first column."
    
    and Henry himself describes the construction of that table as-
    "Tabular log = log sin (30 deg) + log cos Moon's apparent altitude +
    [prop-log] of Moon's altitude corrections."
    
    So built into table I (for the Moon, and similarly into tables II or III
    for star or Sun) is its own "Moon's altitude correction" (for refraction
    and parallax), which the table can deduce from its knowledge of the
    altitude, and which the user has no means of tinkering with. That's what
    prevents him from applying corrections to refraction for extreme climatic
    conditions, or allowing for the parallax of Venus or Mars.
    
    I wonder if Henry is convinced. As I said, it's not a big deal.
    
    =================
    Henry continues-
    
    One cavalier treatment, not previously mentioned, is that
    >Arnold advocates a standardized observed altitude correction to obtain
    >the apparent altitude; under a Rule III, he advocates, across the board
    >...
    >
    >"To the moon's observed altitude, add 12', if the lower limb be taken,
    >but if the upper limb be taken, subtract 20'. to the observed altitude of
    >
    >the sun's lower limb add 12', and from the star's observed altitude
    >subtract 4', and you will have their apparent altitudes." Of course,
    >we know this to be technically incorrect - perhaps it is simply a
    >reflection of the often expressed opinion that an error of a few
    >minutes of arch in altitude does not materially affect the result in
    >clearing the distance.
    
    Comment-
    
    This seems a crude approach, but it's a perfectly valid approximation for
    the purpose of correcting lunars, because the altitude corrections only
    vary slowly with altitude itself, and because it's those corrections,
    rather than the altitudes, that are so vitally important for clearing the
    lunar distance.
    
    ==============
    
    I should point out that anyone who, like me, tries to get to the bottom of
    Mendoza's approximate method, and Arnold's method, will find a description
    in Cotter's "A history of nautical astronomy" , pages 227 to 231, of
    Merrifield's method. This is very similar to Mendoza and Arnold, and
    Cotter's analysis showed the way that the trig was done. Without that, I
    couldn't have tackled Mendoza's or Arnold's recipes. Taking Cotter's
    description at face value, the main difference seems to be this- Merrifield
    simply disregards the final correction term, provided by Arnold's Table
    VII.
    
    George.
    
    ===================================================
    
    "A short Method of Correcting the Apparent Distance of the Moon from the
    Sun or Star.
    
    Invented by the Author.
    
    Rule
    
    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 altitude, call
    the first remainder.
    
    The difference between the half sum and the moon's apparent altitude, call
    the second remainder.
    
    Set down-
    
    The sine of the apparent distance in two columns
    The secant of the half sum also in both columns
    The cosecant of the first remainder in the first column
    And the cosecant of the second remainder in the second column.
    
    Enter Table I with the Moon's apparent altitude and horizontal
    parallax, and take out the corresponding logarithm, which place
    in the first column.
    
    Enter Table II, if a star is used, or table III, if the sun is used, and
    take out the corresponding logarithm, which place in the second column.
    
    The sum of these four logarithms, rejecting the 10's in the indexes, in the
    first column. will give a proportional logarithm of the first correction.
    
    And the sum of the four logarithms in the second column, rejecting the 10's
    in the indexes, will be the proportional logarithm of the second
    correction.
    
    Add to the apparent distance the first correction and the correction of the
    sun or star's altitude, and subtract the sum of the second correction, and
    the correction of moon's altitude will be the corrected distance.
    
    Then enter Table VII, with the corrected distance at the top, with the
    difference of the first correction, and the correction of the moon's
    altitude in the left side column, and also in said table with the
    correction of the moon's altitude in the left side column, and take out two
    corresponding numbers. The difference between the two numbers is to be
    added to the corrected distance when less than 90 degrees, or subtracted if
    above 90 degrees."
    
    Henry Halboth adds the following notes about the tables-
    
    "Relevant included tables".
    
    Table I = A table of logarithms against top entries of Moon's Apparent
    Altitude and with side entries of Moon's Horizontal Parallax.
    
    This table is constructed/calculated as follows...
    
    Tabular log = log sin (30 deg) + log cos Moon's apparent altitude +
    [prop-log] of Moon's altitude corrections.
    
    Table II = A table of logarithms agains Star's Apparent Altitude
    
    This table is constructed/calculated as follows...
    
    Tabular log = log sine (30 deg) + log cos Star's apparent altitude +
    [prop-log] of Star's altitude correction.
    
    Table III = A table of logarithms against Sun's Apparent Altitude.
    
    This table is constructed/calculated as follows...
    
    Tabular log = log sine (30 deg) + log cos Sun's apparent altitude +
    [prop-log] of Sun's altitude correction.
    
    Table VII = A table of corrections against corrected distances across top
    and the difference between the first correction and moon's altitude
    correction or the moon's altitude correction alone as side entries. This
    table provides a third correction to the Distance for parallax and
    refraction.- it is essentially Norie's Table XXXV, presented in a slightly
    different manner.
    
    There are other tables included which are essentially as contained in any
    navigational epitome, i.e., altitude corrections for the Moon, Sun, and
    Stars, etc., which are unnecessary of comment at this time."
    
    ================================================================
    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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