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    Approximate methods for clearing the Lunar distances - some details; corrected version
    From: Jan Kalivoda
    Date: 2003 Apr 13, 18:16 +0200

    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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