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    Re: Classification of the methods for clearing the Lunar Distances
    From: Arthur Pearson
    Date: 2003 Apr 7, 23:11 -0400

    Many thanks for this very thorough treatise.  This is the first time I
    have been able to picture the use of perpendiculars between M and S and
    ms as a basis for an approximate solution. I will need to draw some
    diagrams and reread section 3 a few times. I would love to better
    understand the "long line of always diminishing trigonometrical terms of
    corrections" that seem to be the essence of the approximate approach,
    but I have a long way to go.  Any diagrams you may have illustrating
    this concept I would happily post on the lunar distance website so
    others as curious as I can get a better understanding. Any format would
    do, including scanned handwritten papers.
    -----Original Message-----
    From: Navigation Mailing List
    [mailto:NAVIGATION-L{at}LISTSERV.WEBKAHUNA.COM] On Behalf Of Jan Kalivoda
    Sent: Monday, April 07, 2003 3:51 PM
    Subject: Classification of the methods for clearing the Lunar Distances
    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) 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
    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 19th century!
    - inspection tables (compare HO 214, 218, 240, 229 and ancient Ball's
    tables, firstly edited in 1907)
    - "short" methods (compare Ageton's method in HO 211, Dreisenstock's
    method in HO 208, Smart, Ogura, Aquino 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 Nautical Almanac.
    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 (further L.D.'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 !
    L.D.'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 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
    her 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 L.D.) and MS (
    true=cleared L.D.) cross each o!
    ther at the common point X. But the sections mM and sS are very short
    (half 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 L.D.) and vice versa, we can trigonometrically deduce
    approximate equation permitting to reduce ("clear") the apparent L.D. to
    the true L.D. (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 L.D., 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
    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
    But from the beginning of the 19th century seamen were not left alone
    with these approximate methods. Many proposals of simpler procedures
    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 practise, as introduced from 1810:
    D = d - HP sin s cosec d + HP sin m cot d + MYSTERY
    The navigator only computed the two first corrections by logarithms of
    trigonometrical functions to 4 figures and by proportional logarithms
    originally tabulated by Maskelyne for interpolating the tabulated L.D.'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 lunar
    distance observed.
    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 firstly in 1810 and 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 1820). In
    1851 the 42th edition appeared! 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 40-60 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 method.
    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 practise 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 L.D.'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 the Thomson's method - the sign of author's growing
    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 French astronomer 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 (Newcombe'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 survive, at le!
    ast for checking the chronometers.
    Maybe the method of Bruce Stark is the last method invented in this
    class, but I don't know anything about it.
    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 straightforward
    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 -
    Mackay improved this form by using versines instead of cosines in 1819,
    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 courses.
    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]
    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 requesting 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 surely the frequent case).
    Thank you for your corrections and supplements.
    Jan Kalivoda

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