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    Classification of the methods for clearing the Lunar Distances - corrected version
    From: Jan Kalivoda
    Date: 2003 Apr 13, 14:05 +0200

    Thanks to numerous comments and corrections from several list members, I can 
    send a corrected version of my article with same subject from 7th April into 
    the list. I supposed that it would be appropriate.
    Jan Kalivoda
    As you all know very well, the key step in the finding the GMT by lunar 
    distances is to compare the distance measured by the sextant or the repeating 
    circle with the values tabulated in almanacs (after 1767, when the first 
    volume of the Nautical Almanac was published by Nevil Maskelyne; in ten 
    previous years another interesting method was used by a handful of informed 
    navigators - rather navigating astronomers).
    But the measured distance is "dirtied" by the effects of refraction and 
    parallax on the altitudes of both bodies (although the parallax of the other 
    body was often neglected, even in the case of Sun or Venus; stars have 
    absolutely negligible daily parallax, of course). Therefore this measured 
    lunar distance must be "cleared", i.e. reduced to the theoretical value that 
    would be observed from the Earth's centre in vacuum and only then it can be 
    compared with tabulated values of the Almanac so as to obtain the GMT.
    This "clearing" is difficult part of "lunars" and about hundred procedures 
    were devised for this purpose, beginning from 1750/1759 when the Frenchman 
    Lacaille (La Caille, known by creating several names for faint southern 
    constellations, too, e.g. Circinus, Fornax, Horologium etc.) proposed the 
    first one applicable on the basis of studies of his countryman Jean Morin, 
    who had analyzed the problem in 1633.
    Maybe it would be of some profit to classify these methods according to their 
    principles. I will try it as a modest additamentum to the valuable book of 
    Charles Cotter "A history of nautical astronomy", London 1968, which pays 
    little attention to older and the most important and renowned methods from 
    the times before 1850, when the "lunars" were at their best.
    We can distinguish four classes of these methods, which are remotely similar 
    to the classes of the methods for reducing sights by "Marcq St Hilaire 
    (intercept) method", the only method for using celestial lines of position 
    surviving in today's navigation. These are in the order of their increasing 
    length, difficulty and logical clearness and beauty (in my eyes):
    - software solutions; quite common now and not unknown in the first half of the 19th century!
    - inspection tables (compare HO 214, 218, 240, 229 and ancient Ball's tables, 
    firstly edited in 1907 - compare HO 208)
    - "short" methods (compare Ageton's method in HO 211, Dreisenstock's method in 
    HO 208, Aquino (HO 200), Smart, Ogura, etc.; in these methods short tables 
    with auxiliary values are provided that are combined to obtain the end 
    result; these tables were much less bulky and expensive than the inspection 
    tables, but their use was more difficult and time-consuming)
    - rigorous solutions (compare cosine-haversine formula)
    1. Software solutions
    Yes, the third mechanical computer of human history (preceded by Descartes' 
    and Leibniz' machines) was created for computing the corrections of lunar 
    distances. Its designer was Charles Babbage (1792-1871), who projected this 
    programmable mechanical device together with Byron's daughter Ada after 1822. 
    The machine was programmed by predecessors of punched cards. Its prototype 
    survived to our days, but did never function.
    2. Inspection tables for clearing lunar distances
    The plural is not appropriate - only one such work appeared. It was "Tables 
    for correcting the apparent distance of the Moon and a Star from the Effects 
    of Refraction and Parallax", Cambridge 1772, in folio. It is commonly cited 
    as "Cambridge Tables", or sometimes as "Shepherd's Tables" (A.Shepherd was 
    the author of the preface, but took no part in computing the tables). They 
    were computed and edited in the first spell of enthusiasm for lunars, after 
    Tobias Mayer's lunar tables were edited in 1770 and used even before in the 
    manuscript form by Maskelyne for editing the first volumes of the Nautical 
    Cambridge Tables were an incredible deed. After 4 pages of foreword and 7 
    pages of instructions 1104 (thousand hundred four) pages follow with up to 
    370 corrections on each page, together cca 300000 values. Corrections were 
    computed and arranged for each degree of lunar distance from 10 to 120 
    degrees. Each degree of distance occupied 3-14 pages. For each degree of 
    distance all possible combinations of Sun's and Moon's altitudes (stepped by 
    one degree) were evaluated and the corrections of apparent lunar distances 
    (onwards LD's) for Moon's horizontal parallax of 53 arc-minutes and the mean 
    refraction were given. Other two table columns gave the corrections for the 
    actual Moon's horizontal parallax and the actual air temperature and 
    pressure. Of course, triple interpolation was needed, but second differences 
    were negligible, rarely exceeding 3 arc-seconds. Small table for correcting 
    for horizontal parallax of the Sun (9 arc-seconds) was given. Planets were 
    not yet used for LD!
    's in that time.
    The head of the working group of calculators was probably Israel Lyons, who 
    prepared the clever method of computations (one of "short" methods, mentioned 
    below), too. After editing this giant work, he took part in Phipps' polar 
    expedition in 1773, but died at home in 1775 in the age of 36 years.
    Of course, these folio tables were too bulky, cumbersome and costly to gain 
    any popularity at sea. Very small number of their copies have survived to our 
    days in great libraries.
    3. "Short" or "approximate" methods
    Imagine the triangle in the sky with the vertices Z - zenith, S - true 
    Sun/star and M - true Moon. And another triangle with wertices Z - zenith, s 
    - apparent=observed Sun/star and m - apparent=observed Moon. The two 
    triangles have the common vertex (and angle) Z and their two sides (zenith 
    distances of the four bodies mentioned!) crossing at Z and perpendicular to 
    the horizon coincide for the most part of them: s lies above S, as the daily 
    parallax (which always lowers the apparent=observed body below 
    true=supposed-to-be body for an observer on Earth's surface) of the Sun or 
    planet (not mentioning the stars) is always much smaller then the effect of 
    refraction (which always raises the apparent body above true body). On the 
    contrary, m lies below M, as its great daily parallax is always greater then 
    the effect of refraction. As a result, the third sides (apparent and true 
    lunar distance!) of both triangles, ms (apparent=observed LD) and MS ( 
    true=cleared LD) cross each other!
     at the common point X. But the sections mM and sS are very short (one degree 
    at most, but mostly shorter), which is essential for further procedures.
    Therefore if we drop perpendiculars from the points M and S to the side ms 
    (apparent LD), we can trigonometrically deduce approximate equation 
    permitting to reduce ("clear") the apparent LD to the true LD (Here you can 
    see a very remote similarity with Ageton's and other methods for resolving 
    the nautical triangle; but these are not approximate in any degree, only 
    their use of perpendiculars to triangle sides is somewhat similar.)
    (M,S,m,s are meant as centres of bodies - the limbs are measured, of course, 
    but applying the corrections for the semidiameters of bodies, one obtains the 
    values for centres. I neglect all three efects of ellipsoidal earth's shape 
    on clearing LD, too; they can make a maximal error of 13 arc-seconds in the 
    true distance cleared, when neglected.)
    The final approximate formula can be confirmed directly by the calculus 
    (Taylor's polynoms), too, but the spherical trigonometry alone can find the 
    long line of always diminishing trigonometrical terms of corrections allowing 
    for effects of parallax, refraction and their combinations on a measured 
    lunar distance. 10 (ten) terms were sometimes used for calculation! This 
    formula is called "approximate", as it is not derived strictly, but only in 
    gradually approaching steps and terms; but when sufficient number of terms is 
    included, its accuracy leaves nothing open.
    The first methods of this kind were the methode of Lacaille (1759) and Lyons 
    (1766); both were mentioned above. Another was Witchell's method from 1772 
    (the "fourth method" of Bowditch). But their formulas were too complicated 
    for seaman's everyday use, therefore Dunthorne's and Borda's rigorous methods 
    (see below in the fourth chapter) were more popular then.
    But from the beginning of the 19th century seamen were not left alone with the 
    approximate equation. Many proposals of simpler procedures appeared:
    D,d - true and apparent=observed lunar distances
    M,m = true and apparent=observed ALTITUDES of the Moon (NOT its centres as above!)
    S,s = true and apparent=observed ALTITUDES of the Sun/star (NOT its centres as above!)
    HP = horizontal parallax of the Moon
    The formula for the sea practice, as introduced from 1810:
    D = d - HP sin s cosec d + HP sin m cotg d + MYSTERY
    The navigator computed only the two first corrections by logarithms of 
    trigonometrical functions to 4 figures and by proportional logarithms 
    originally tabulated by Maskelyne for interpolating the tabulated LD's in the 
    Nautical Almanac; that were two greatest terms of Moon's parallax in the 
    "approximate" equation, mentioned above.
    And the MYSTERY was the "third correction", tabulated according to the values 
    of Moon's and Sun's/star's altitudes observed and of the lunar distance 
    The main difference between various methods of this numerous class was, how 
    many secondary terms (from these remaining eight terms in the "approximate" 
    equation) were taken into account; the authors seldom stated these details 
    and published their tables as they were - sailor, take it or leave it!
    The second difference between various tables was their step, of course, and 
    consequently the amount of the interpolation needed. Several were even 
    arranged as nomograms, in a graphical form.
    The first table of this kind (after two unpublished or unnoticed predecessors) 
    was the publication of merchant master Elford from Charleston, which appeared 
    in 1810 for the first time and which was several times reedited and many 
    times stolen by other "authors" up to the end of 19th century. Elford's table 
    of the "third correction" included only two greatest terms of refraction, 
    leaving other six smaller refraction and parallax terms aside.
    The same value is given in the "Set of linear lables for correcting the 
    apparent Distance of the Moon from the Sun or a fixed Star for the effect of 
    Refraction", edited by well-known J.W.Norie in 1815 in London. That work 
    contained 24 nomograms, from which the "third correction" could be taken 
    without any interpolation with the precision of 2 arc-seconds. This set was 
    popular, but never edited again, as original engravings of nomograms were 
    difficult to obtain. So was Norie protected from thiefs that irritated Elford 
    so much and so often. But sailors had to leave this tool.
    But the most prominent author of the tables in this class was David Thomson, 
    who published the workhorse of British navigators in the first half of the 
    19the century: "Lunar and Horary Tables for new and concise Methods of 
    performing the Calculations necessary for ascertaining the Longitude by Lunar 
    Observations or Chronometers..." (London 1824). In 1851 the 42th edition 
    appeared, in 1880 the 67th edition! (Thanks to Bruce Stark for this 
    information.) And again was his main table accepted (i.e. stolen) into many 
    other nautical tables collections.
    It was an ace of nautical tools in that time. Firstly, it gave on 51 pages (so 
    that no interpolation was necessary) the value of the mysterious "third 
    correction", allowing (as opposed to Elford and Norie and others) for further 
    smaller terms of the complete approximate formula. It brought the improvement 
    of 90 arc-seconds to the precision of corrections in some (not very frequent) 
    unfavourable situations. A small table was given for reducing the parallax 
    effect of the other body used.
    Secondly, the Thomson's table set included auxiliary tables for computing the 
    first two Moon-parallax corrections of the simplified formula mentioned 
    earlier that the seaman had to resolve directly. Taken together, Thomson's 
    tables permitted the shortest method for clearing lunar distance ever 
    contrived - it was shorter than reducing the Sumner line by cosine-haversine 
    And many other useful tables were included, e.g. for resolving "time sights" 
    (i.e. measuring altitudes of celestial bodies for computing their local hour 
    angle to be compared with the chronometer time or "lunar" time for "finding" 
    the longitude) by cosine-haversine method, tables for finding azimuths of 
    celestial bodies and so on.
    David Thomson went the long route from the ordinary soldier and seaman to the 
    merchant master. He died in 1834 in Mauritius as a storekeeper, unknown and 
    enigmatic personality. He never specified the method of computing his main 
    table of the "third correction". It was guessed that he had to compute 30000 
    lunar distances directly and to interpolate another 50000 values so as to 
    construct this table. His results were proved to be independent of "Cambridge 
    Tables" and are better than theirs in the average. But his caginess about his 
    computing method prevented his table from entering into the navigation 
    courses and navigation practice aboard navy ships, which were not insured.
    The Thomson's method and tables (after being simplified) were taken over by 
    Bowditch as his "second method" for clearing the LD's, as Bowditch states 
    expressly (he spells him "Thompson", but in my other sources the name always 
    sounds "Thomson") The "first method" and "third method" of Bowditch, which 
    were devised by himself, and his "fourth method", improved from Witchell's 
    procedure (see above), were "short/approximate" methods, too, but they were 
    rather obsolescent after 1810, as their lenght and greater number of 
    necessary arithmetical operations in comparison with Thomson's "second 
    method" prove in Bowditch's examples. (The "first method" stood in the 
    appendix in the first Bowditch's editions and only later he shifted it into 
    the main text to the head before Thomson's method - the sign of author's 
    growing self-confidence.)
    Of course, in the second half of the 19th century some other 
    "short/approximate" methods appeared that didn't resemble the Elford/Thomson 
    solution. Some are mentioned in Cotter's book. Another was the method of the 
    American astronomer William Chauvenet that replaced all other older methods 
    in "American Practical Navigator" in the year 1888 (the pertinent pages were 
    scanned and published on the web by Dan Allen for this group). This method, 
    in contrast to the all mentioned above, was capable to take into account ALL 
    effects of ellipsoidal Earth's shape and temperature/barometric corrections 
    of mean refraction values. In competition with widely used chronometers and 
    owing to very precise lunar positions in almanacs from 1880 (Newcomb's superb 
    equations of planetary and lunar motions began then to be used for 
    ephemerides), the editors supposed in this year that "lunars" should be given 
    a more precise, although more laborious method in the "American Practical 
    Navigator" to survi!
    ve, at least for checking the chronometers.
    4. Rigorous methods for clearing the lunar distances
    The most logical class comes the last. Take the triangle zenith - true 
    Sun/star - true Moon and the second triangle zenith - apparent Sun/star - 
    apparent Moon once more. They have the common vertex and angle at zenith. 
    This permits to compare the basic trigonometric equations for both spherical 
    triangles and deduce various strict trigonometric formulas for finding the 
    true lunar distance, when apparent=observed lunar distance and 
    apparent=observed and true altitudes of both bodies used are known (we can 
    obtain the true altitudes from apparent=observed altitudes very quickly by 
    allowing for refractions and parallaxes).
    So again:
    D,d - true and apparent lunar distances
    M,m = true and apparent altitudes of the Moon
    S,s = true and apparent altitudes of the Sun/star
    A = auxiliary value
    Two most popular methods of this class were Dunthorne's and Borda's method. I 
    won't write out their deduction, only the final forms:
    Dunthorne (1766): cos D = cos(M-S) + cos M cos S sec m sec s [cos d - cos(m-s)]
    Mackay improved this form by using versines instead of cosines in 1793 (he 
    edited tables for this purpose in 1809), removing the small incovenience of 
    changing the sign of cosine at 90 degrees by this substitution. Young's 
    formula from 1856 is very similar to the original Dunthorne's form.
    The Dunthorne's method was very popular in German speaking countries and in 
    Scandinavia up to the beginning of 20th century, at least in navigation 
    Borda (1778):
     cos A squared = cos M cos S sec m sec s cos[(m+s+d)/2] cos[(m+s-d)/2]
     sin D/2 squared = sin[A + (M+S)/2] sin[A - (M+S)/2]
    Another slightly simpler forms of these equations are published. In spite of 
    it, I cannot understand, why this cumbersome method gained such popularity. 
    But it was widely used in France and other Romance speaking countries and 
    many successors devised similar formulas: Delambre, Krafft (a bulky volume of 
    auxiliary tables in 600 pages were collected for that method by Mendoza del 
    Rios in 1801) and others.
    In all these equations the term (cos M cos S sec m sec s) returns again and 
    again. It was called "logarithmic difference" and tabulated in an inspection 
    table according to the apparent altitudes of the Moon and of the other body. 
    An error of some 3-5 arc-seconds arose from its use, but this was considered 
    tolerable before 1850.
    The great disadvantage of all rigorous methods was that they requested the use 
    of logarithms to 6 figures (and some theoreticians frowned at it, vainly 
    requiring the use of the logarithms to no fewer than 7 figures), whereas the 
    approximate methods were quite satisfied with logarithms to 4 figures with 
    the same accuracy. The difference in difficulty of computations is manifest.
    On the other side, all rigorous methods were capable of all three corrections 
    for ellipsoidal Earth's shape and of corrections for the actual thermometer 
    and barometer values (effects on the mean refraction), whereas these 
    corrections are difficult or impossible to use in the most approximate 
    methods (except from tedious Chauvenet's method, see above). And each step of 
    calculation was under the full control of navigator in rigorous methods, 
    where one can be sure that if logarithmic tables are correct (which could be 
    guaranteed almost surely even in the 18th century), the result depends only 
    on navigator's sextant, hand and mind. Approximate methods with their 
    mysterious tables required a bit fatalistic seaman (which was certainly the 
    frequent case).
    The last rigorous method invented is the very ingenious procedure of Bruce 
    Star from the last years, which overcomes the disadvantages of rigorous 
    methods and retains their advantages. Unfortunately, I don't know much about 
    it, hence I cannot give any details.
    Thank you for your further corrections and supplements.
    Jan Kalivoda

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