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    Re: Easy Lunars in 1790
    From: Frank Reed CT
    Date: 2006 Apr 27, 22:31 EDT

    Ken Muldrew, you wrote:
    "One of the real gems  among the gold mine of 18th century navigation
    documents that everyone has been  downloading is Margetts' Longitude Tables by George
    Margetts, published in 1790.  "Tables" was probably a poor choice for a title
    as the collection is really a  series of graphs allowing one to clear a lunar
    distance by interpolating the  necessary corrections."
    Yes, they're wonderful, aren't they? Speaking of  peculiar terminology,
    you've probably noticed that Margetts (also a few years  later, Norie) uses
    "formulae" to refer to fill-in worksheets or work forms  (formulae --> forms, I
    You noted that you could clear a  lunar in 5 minutes with Margetts tables
    while Witchell's method required an  hour. I gotta say, I don't think this would
    be a fair comparison for most  people. Yes, you can get some speed with those
    look-up graphs in Margetts's  book, but you can also become efficient at using
    Witchell's method. The total  process of clearing a lunar and working the
    time sight via Witchell, in my  opinion, takes about 25 minutes. The same total
    time with Margetts might require  five minutes less. It's a savings, yes, but
    surely not an order of magnitude  improvement.
    " In short, the "problem" of clearing lunar  distances was just as
    fast and easy to solve in 1790 (if you had Margetts'  book) as it is in
    2006 using a pocket calculator."
    Over the past 240  odd years, time and again commentators on navigation have
    assumed that lunars  would be more popular if only there was a different way
    of clearing them. I  think this is fundamentally mistaken. Lunars are not
    difficult mathematically.  There are numerous methods. Some are a bit more tedious
    than others, but they  all have a lot in common and none of them is really
    difficult from a  calculational standpoint. It's true that deriving lunars
    calculations and  analyzing them involves some mathematical difficulty, but that's
    another matter  entirely.
    And you wrote:
    "Each of these lines is for a horizontal  parallax of 53'. There
    are also dotted lines on the graphs (maybe 10-20 per  graph) that give the
    parallactic correction for a horizontal parallax of 62'  over 53'."
    This is the really clever part of the diagrams. At the most  complete level,
    clearing any lunar distance is a function of the following  variables:
    d, the observed lunar distance
    h_s, the Sun's or other object's  altitude
    h_m, the Moon's altitude
    P, the Moon's parallax
    p, the Sun's  or other object's parallax
    T_P, the temperature/pressure factor and
    L,  observer's latitude (for oblateness).
    If we ignore L, p, and T_P, as Margetts  does, we've only got four variables.
    It's not too difficult to use graphic plots  to read off a quantity that
    depends on three variables, but it's tricky when you  get to four. Margetts's
    procedure handles it beautifully.
    You also  wrote:
    "One only needs to calculate the corner
    cosines to find out the  relative contributions of correction for each
    value of the moon's altitude  (see Frank Reed's posts in the archives on
    "Easy Lunars" for a full  explanation of this operation). Basically, one
    needs to solve the following  two  equations:
    where  dM and dS are the corrections for the moon and star(sun)
    respectively, s_alt  is the altitude of the star(sun), m_alt is the
    altitude of the moon, and d is  the distance between them."
    I know you're just giving this as an example  of how it might be calculated,
    but, for what it's worth, I think Margetts  probably used Shepherd's Tables,
    instead of a series approach.
    And you  asked:
    "So the question is, why is the history of navigation utterly silent  on this
    brilliant method to clear lunar distances? In 1790, clearing the lunar
    distance could have been made the most trivial part of finding one's longitude,
    yet navigators persisted in flipping through log tables and following arcane
    recipes, torturing themselves to clear the distance (well, not exactly torture,
    but certainly an unpleasant half hour even at the best of times)."
    I  don't think it was all that painful, as I've said above, but it's still an
     excellent question: why isn't this book better known? I've got a few
    thoughts on  this...
    Fos starters, we're talking about a commercial product, in  competition with
    many other products. Margetts made these tables to sell. If  they performed
    poorly in the marketplace, there could be a number of  explanations. Maybe he
    priced them too high. Maybe they were perceived as a poor  value considering
    that they were useful for only one topic in navigation. But  they certainly
    weren't a total flop commercially. I note that Edmund Blunt was  selling them in
    1817. In an advertisement for his very successful store in  Manhattan, he lists
    navigation books for sale in this order, "Rio's Tables for  Navigation and
    Nautical Astronomy [Mendoza Rios]; Bowditch's Practical  Navigator; Lyon's Tables
    for working the longitude at sea, being the shortest  method used; Margett's
    Tables; Mackay's Longitude, 2 vols." along with other  non-navigational
    publications. So they were available in New York decades after  their first
    publication, though that doesn't prove they sold well. It's also  worth noticing that
    graphs like his were almost impossible to copy. This was  presumably good for
    Margetts's wallet since it made piracy more difficult but  perhaps not so good
    for his fame in the long haul since they couldn't be  re-published easily by
    compilers of other navigation books, like Nathaniel  Bowditch.
    Also, there was a real bias against graphical methods in this  era. Why? I
    don't know. As you note, this bias seems irrational to us. But maybe  that just
    means we haven't gotten inside the heads of those folks back then  yet.
    Finally, there is the purely technical matter of accuracy, which I  consider
    least important in this case. Was anyone bothered by the fact that the
    refraction couldn't be corrected for temperature and pressure? The graphs also
    ignore the Sun's parallax. That's fine if you're using the stars, and it's a
    small correction anyway, but if most practicing navigators used the Sun
    considerably more often than the stars for lunars, this might have seemed like a  point
    against Mr. Margetts and his "tables".
    42.0N 87.7W, or  41.4N 72.1W.

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