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    What were the Lunar Distances for?
    From: Jan Kalivoda
    Date: 2002 Dec 26, 17:16 +0100

    I am sorry that I touch this topic again. I had thoroughly studied your 
    discussion of the past months about the matter and, thanks to George 
    Huxtable, Cotter and some older German-written literature, I am informed 
    about this navigational method and its background. Thank you very much, Mr. 
    Huxtable, for your effort, enthusiasm and above all for your wit and 
    But I should repeat the question: What were the L.D. useful for?
    Let us consider the limit of their inherent errors. We can put pure personal 
    errors of observation aside in a forlorn hope that they will be averaged in 
    greater sets of shots.
    But take the instrument error. When measuring lunar distance, we must guess 
    its value at 1 minute of arc for sextants from the year 1800 and at half a 
    minute of arc for the year 1860 and for our times.
    The second important error was the one of the Moon tables. Up to 1810 the 
    original table of Tobias Mayer (edited by Maskelyne as "Tabulae motuum Solis 
    et Lunae" in 1770) was being used for computing the lunar distances in 
    almanacs. Its internal accuracy for Moon's positions in ecliptical 
    coordinates was 1 minute of arc, as Mayer himself stated. (This was the cause 
    or pretence for the fact, that Mayer's widow obtained "only" 3000 pounds from 
    the Board of Longitude, in comparison to 24000 pounds for Harrison.)
    Between 1810 and 1860, Laplace, B�rg, Burckhart, Pierce and Hansed improved 
    this accuracy to 25 seconds of arc. Only after 1883, when all great almanacs 
    adopted Newcomb's corrections for calculations, the accuracy of Moon's tables 
    rised to 1 (one) second of arc. But it was too late then, "lunars" were as 
    dead as Julius Caesar already (as Lecky in his "Wrinkles" said. Today, the 
    accuracy of Almanac is not an issue, but this is compensated for by 
    difficulties of observing distances from small boats and related errors.)
    Combining these two factors together, we must cope with the maximum error (not 
    along the Gauss curve, but with the quadrate distribution, thus much more 
    frequent) of 2 minutes of arc in the lunar distance for the year 1800 and 1 
    minute for 1860. This error created an error of 4/2 minutes in the GMT 
    obtained and 60/30 minutes of longitude, respectively. (Sometimes twice more, 
    according to the effect of the change of paralax near the zenith, as George 
    Huxtable very cleverly emphasizes.)
    What could a sailor achieve on the basis of this piece of information? He 
    could not rate his chronometer by lunars regularly, as the rate acquired 
    would be too irregular (if intervals had been short) or not reliable (if 
    intervals had been long, according to the varying temperature influence on 
    the chronometer rate) or both (between). And he could not check his D.R. 
    navigation continuously, as the inaccuracy of lunars (30 - 60 minutes of 
    longitude) equals 10 - 30 % of daily run with sailing ships and the first 
    steamers (we should omit the best runs of the best clippers).
    I can imagine only two modes of using lunars for navigation in the past - 
    firstly, the check of D.R. position before a landfall when finishing a long 
    deep-see passage. But even then, owing to inaccuracy of lunars, so called 
    "easting/westing" from the target and the final latitude sailing would be 
    necessary, only the initial amount of this deliberate longitude deflection 
    could be smaller.
    And secondly, one could roughly check his lone chronometr by lunars - when the 
    chronometer time fell within four (two) minutes of time found by lunars, the 
    chronometer could perhaps be relied upon. Otherwise, it was to be neglected 
    and latitude sailing (see above) took place.
    Even so, I wonder if many captains in the merchant sailing fleet used the lunars regularly.
    Maybe, I will be advised differently by you. I hope so and thank you in advance.
    Jan Kalivoda

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