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    Re: Approximate methods for clearing the Lunar distances - some details; corrected version
    From: Arthur Pearson
    Date: 2003 Apr 19, 22:53 -0400

    Jan,
    
    I have re-read this posting and continue to find it fascinating.  It
    paints a vivid picture the extraordinary efforts expended to find an
    accurate method of clearing the distance that was also convenient for
    use by the average sea captain. The derivation of these formulae is
    brilliant in itself. The ingenuity required to transform them in ways
    amenable to tabular solution is stunning.  The determination to grind
    through 30,000 LD solutions and another 50,000 interpolations to
    actually produce a table is beyond imagination.  Sending a man to the
    moon offers nothing more impressive than the achievement of making
    lunars a practical tool.
    
    Because I need to condense things to understand them, I have amended my
    diagram to illustrate all the variables you mention and copied over the
    polynom, the substitutions, and the approximate formula just below it.
    It is still at
    http://members.verizon.net/~vze3nfrm/Nav_L_Graphs/ApproxLD.JPG. Please
    let me know if you see any mistakes in my transcription of formulae or
    placement of variables. Needless to say, links to both branches of this
    thread are now on the Nav-L section of www.LD-DEADLINK-com.
    
    My only remaining questions: What exactly is a polynom and where in the
    derivation of formulae are those perpendiculars used?
    
    Thanks again,
    Arthur
    
    
    -----Original Message-----
    From: Navigation Mailing List
    [mailto:NAVIGATION-L@LISTSERV.WEBKAHUNA.COM] On Behalf Of Jan Kalivoda
    Sent: Sunday, April 13, 2003 12:16 PM
    To: NAVIGATION-L@LISTSERV.WEBKAHUNA.COM
    Subject: Approximate methods for clearing the Lunar distances - some
    details; corrected version
    
    I am sorry, but when posting this article for the first time, symbols of
    halving and of squaring were distorted in formulas on the web. Therefore
    I send it for the second time. Erase the first version, please and take
    the symbols "P2", "sin2" as "P squared", "sin squared" onwards. The
    original article follows:
    
    =====================
    
    
    In my previous article on classification of methods of clearing the
    lunar distances, I didn't want to burden its text by greater details.
    But some of you replied that they would like to read more about the
    approximate methods for clearing the lunar distances ("lunars", LD's)
    and about the deduction of their basic formula. Therefore I can perhaps
    try to give more details now - only for the interested guys.
    
    Meanwhile Arthur Pearson prepared a nice picture for this purpose on his
    delightful website devoted to "lunars" (http://www.ld-DEADLINK-com/).
    I thank him very much, it was beyond my graphic abilities. Use this
    picture as the reference, as I will do onward:
    
    http://members.verizon.net/~vze3nfrm/Nav_L_Graphs/ApproxLD.JPG
    
    (In this picture and in the following text you can always take a star or
    a planet for the Sun ! )
    
    
    Consider the spherical triangle: Zenith - true Moon - true Sun; and the
    second one: Zenith - apparent Moon - apparent Sun. They have the common
    angle Z'. Therefore we can deduce various strict fomulas for the true LD
    between the true Sun and the true Moon, using the apparent LD and four
    apparent and true zenith distances in these two triangles. This is the
    way of rigorous methods.
    
    But as the distances apparent Sun - true Sun and apparent Moon - true
    Moon are very small in comparison to other elements of both triangles,
    we can also search only for small corrections that can be algebraically
    added to the apparent=observed LD to obtain the true LD. This is the way
    of approximate methods.
    
    These corrections can be found by two procedures: by the calculus or by
    using the spherical trigonometry of perpendiculars dropped between the
    vertices and sides of two triangles mentioned (see the lower part of the
    picture). The end formula CAN be the same in both cases (as always in
    the nautical astronomy, several equivalent solutions are possible,
    according to the aim of their inventors).
    
    The trigonometrical deduction would require some hundred lines; so I
    confine myself to the calculus procedure.
    
    Symbols:
    
    Z - zenith
    Z' - angle at zenith (difference of azimuths of both bodies)
    M, S - true Moon and Sun
    m, s - apparent Moon and Sun
    m', s'  - angles at m, s
    D - true lunar distance (distance MS)
    d - apparent (observed) lunar distance (distance ms)
    A, a - apparent altitudes of the Moon and the Sun over the horizon
    x - the distance Mm, i.e. the difference of the Moon' parallax in A and
    the refraction in A
    y - the distance Ss, i.e. the difference of the Sun's parallax in a and
    the refraction in a
    P, p - HORIZONTAL parallaxes of the Moon and Sun
               (parallaxes in altitudes A, a can be obtained by formulas: P
    cos A , p cos a)
    R, r - refractions in altitudes A, a
    
    
    The starting point is the Taylor's polynom for the small changes x,y of
    two sides Zm, Zs of the spherical triangle Zms, forming the constant
    angle Z':
    
    D = d + y cos s' - x cos m' + 1/2 (y2 sin2 s' + x2 sin2 m') cotg d + x y
    sin s' sin m' cosec d  .....  (an infinite number of smaller terms
    follow that can be and were fully neglected)
    
    
    Of course, this form is unusable at sea. Now three substitutions should
    be made:
    
    y = r nearly, as the maximal parallax of the Sun is 9 arc-seconds, of
    Venus 33 arc-seconds and and of Mars 23 arc-seconds; therefore the
    parallax of the second body can be reduced by a special term at the end
    of procedure and neglecting it at the beginning of arrangements will not
    induce a significant error during the further treatment of the formula;
    Jupiter, Saturn and stars have the daily parallax zero for nautical
    purposes
    
    x = P cos A - R   ; difference of Moon's parallax in its observed
    altitude and of Moon's refraction in its observed altitude!
    
    cos m' = (sin a - sin A cos d) / (cos A sin d)   ; this is the clever
    substitution of Israel Lyons (1766) into the term (x cos m') of polynom,
    resulting from the spherical cosine theorem for the triangle Zms; this
    substitution is essential for the use of the whole formula in this
    branch of approximate methods
    
    
    So after these substitutions and some following bothersome, but
    elementary trigonometric arrangements (let me jump them over now, I can
    send them to you individually, if you like), we obtain the basic formula
    for the most important approximate procedures and their tables for
    clearing the lunar distances from 1810 to the death of lunars:
    
    
    D      =     d     -
    
    (1)           -         P sin a cosec d    +
    
    (2)           +         P sin A cotg d     +
    
    (3)           +         r cos s'     +       R cos m'      +
    
    (4)           +         1/2 (P2 cos2 A sin2 m' cotg d)     -
    
    (5)           -         P R cos A sin2 m' cotg d       +        P r cos
    A sin s' sin m' cosec d      +
    
    (6)           +         1/2 (r2 sin2 s' + R2 sin2 m') cotg d      -
    R r  sin s' sin m' cosec d    -
    
    (7)           -         p sin A cosec d     +    p sin a cotg d
    
    
    
    Remarks:
    
    1,2 - two greatest terms for the Moon's parallax; these two corrections
    were computed by the sailor himself by logarithms to 4 places and
    proportional logarithms; thanks to Lyons' substitution, sailor hadn't to
    be tortured by computing the angle at the apparent Moon and worked only
    with the horizontal parallax of the Moon, with both observed altitudes
    and with the observed distance; Thomson has provided four auxiliary
    tables to speed up these two calculations made by seaman
    
    3 - two greatest terms for both refractions; here the "THIRD CORRECTION"
    begins; ALL THE FOLLOWING (except line 7) was tabulated in ONE table
    according to observed altitudes of both bodies and their observed
    distance; Elford as the inventor of such sort of tables (1810), then
    Norie (1815) and many others tabulated only the value of this line (3),
    leaving all other terms aside
    
    4 - another term for the Moon' paralax; the value of this line was
    tabulated for the first time by Maskelyne in the table No.13 of his
    "Tables Requisite" = table No.35 of Norie's "Epitome of practical
    Navigation" = table No.20 of Bowditch' tables (??? I am not quite sure
    of this last assertion); these tables were used in older approximate
    methods (Lyons, Witchell, Maskelyne, Bowditch) preceding the Elford's
    deed in 1810; they could be used together with Elford's (and similar)
    tables and Norie's nomograms, too
    
    5 - two terms for combined effects of the moon's parallax and both
    refractions
    
    6 - two smaller terms for both refractions
    
    7 - two greatest terms for the parallax of the second body, if any; it
    was given by a special table by Thomson; all smaller terms of this
    parallax remained fully neglected as quite unimportant
    
    
    
    It was my beloved Thomson, who brought this procedure to the state of
    perfection. He never stated his method for computing his pithy table of
    the "third correction" (firstly 1824, the 67th edition in 1880,
    according to the kind information of Bruce Stark), which didn't require
    any interpolation, as its steps were very small. (Compare the table No.
    48 in the Bowditch' old editions (to 1851 at least), which is the
    reprint of Thomson's main table, although it is called "Third correction
    in Lyon's improved method" in the table of contents; but
    Thompson=Thomson is cited in the main text at the "second method" for
    clearing LD's.)
    
    From Thomson's results it has been ascertained that he had included ALL
    terms of the approximate formula given above. He couldn't compute them
    step by step, as it would be an absolutely hopeless effort for one man,
    even for many. From some his remarks (and above all from remarks of
    G.Coleman, the editor of Norie's Epitome after Norie's death, who was
    purportedly in contact with Thomson in his youth) it is probable, that
    Thomson computed only the first two terms from the lines (1) and (2)
    above and then the whole difference between true and apparent distance
    by some rigorous method. And by subtracting those two terms from the
    whole difference found, he obtained the value of remaining terms from
    the lines (3) - (6). The effect of the parallax of the second body from
    the line (7) he tabulated in a special table, as said above.
    
    The improvement of accuracy of Thomson' table in comparison with Elford
    and Norie is most evident, when the lunar distance itself and both
    altitudes are small (and if both altitudes are nearly the same - this
    unfavourable case is seldom in lower latitudes). So for both altitudes
    and lunar distance of 20 degrees, Elford gave the "third correction" of
    22 arc-seconds, Norie the value of 19 arc-seconds, but Thomson (and
    Bowditch) 88 arc-seconds! And this greater value is the true one. The
    importance of smaller terms in the approximate formula (lines 4 - 6) is
    manifest by that.
    
    The general accuracy of Thomson's main table of the "third difference"
    was very high. It has been found out by later trial calculations that
    the error is under 2 arc-seconds in the most cases, and only seldom it
    attains 5-6 arc-seconds. Of course, isolated errors of a much greater
    degree can hide in the tables (which were never recomputed as the
    whole), as in all tables for "approximative" methods. Slocum mentioned
    one such case in the record of his circumnavigation - near the island
    Nukahiva in Marquesas, he ascertained "an error in the important
    logarithm of the tables" during clearing the lunar distance. I suppose
    that he had some table for approximative methods in his hands - he could
    not verify the values of the decadic logarithms themselves aboard, I
    guess. Or does anybody understand this interesting place of Slocum's
    book in a different manner?
    
    But the accuracy of Thomson's tables, as mentioned, must be understood
    only for their default conditions. All tables of the "third difference"
    for clearing LD's by "approximate" methods were liable to the important
    drawback: they were to be computed for the MEAN EQUATORIAL horizontal
    parallax of the Moon (57,5 arc-minutes) and for the MEAN refraction (for
    30 inches of barometric pressure, 55 Fahrenheit degrees of temperature).
    And because their values were the lump of effects of both the parallax
    and the refraction, allowing for the actual Moon's parallax, for the
    actual atmospheric conditions and for the effects of ellipsoidal Earth's
    shape separately was very difficult. I won't enter into these details
    here. But the maximal error created by using only the mean Moon's
    horizontal parallax (57,5 arc-minutes) was 10 arc-seconds at short LD ,
    5 arc-second at LD of 40 degrees and decreasing further with growing LD.
    The combined maximal amount of all three effects of ellipsoidal !
    Earth's shape was 13 arc-seconds. The effect of the anomalous refraction
    could be considerable, of course, above all in the cases, when the LD
    ran vertically to the horizon - the error of 60 arc-seconds is then
    exceptionally possible.
    
    But one had always to expect a MINIMAL error of 30 arc-seconds in
    measuring a lunar distance aboard, even after averaging a set of
    observations. And the tabulated Moon's positions in almanacs (and
    therefore the tabulated lunar distances as well) could be in error of 1
    arc-minute before 1820 and 20 arc-seconds before 1880. So an exaggerated
    accuracy of procedures and auxiliary tables for clearing lunar distances
    was considered meaningless by sailors (not by theoreticians and teachers
    of navigation). In the second half of 19th century the more precise
    forms and tables of approximate methods appeared (Chauvenet after 1850,
    Bolte in 1894 and certainly other ones). But they became so complicated
    in use that they provided no advantage in comparison to rigorous
    methods. So the generic line "(Lyons) - Elford - Thomson - epigons of
    both" prevailed among approximate methods up to the end of lunars.
    
    Of course, Thomson's empirical procedure and table constructed by it
    could not be taken into the official navigation handbooks and courses.
    Thomson's procedure and tables remained an illegitimate darling of
    sailors for 80 years, but they never obtained the status of their
    reputable bride.
    
    
    Jan Kalivoda
    
    
    

       
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