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    Re: Meridional Distances
    From: George Huxtable
    Date: 2006 Jun 29, 15:37 +0100

    Lars Bergman's postings are always perceptive, and he gets things
    right. His latest posting is true to form.
    He wrote-
    | The expression for meridional parts (on a sphere) is
    | a*integral(sec(lat)dlat), from 0 to lat. The solution to this
    | is a*ln(tan(45d+lat/2)) or a*ln(10)*log(tan(45d+lat/2)). With a
    | expressed in arc minutes of a great circle, then
    | a*ln(10)=3437.747'*2.30259=7915.7'. I haven't been able to figure
    | what kind of minutes George uses with his factor 7819, maybe he
    meant to
    | use 7918?
    He is referring to my earlier posting, on 24th June, in a thread "I
    knew where we were, but where are we now?", as follows-
    "...you can convert a
    lat to its equivalent in MP yourself dead easily with a calculator.
    Just halve the lat in degrees (keeping its sign carefully, positive
    North, negative South). Then add 45 degrees (remembering that sign),
    the resulting angle being always positive, between 0 and 90, and find
    the tan of that angle. Now find the log (the ordinary log, to the base
    10) of that result, and multiply it by 7819. It sounds a handful, but
    a calculator will do it with no trouble at all...
    (Note that these values are for a spherical Earth, and to achieve an
    even more precise result, it's possible tinker slightly with the
    conversion of lat to MP to get an even better fit to the Earth's true
    ellipsoidal figure. On that basis are meridional-parts tables, and
    Mercator charts made. We will ignore that tinkering here.)"
    And indeed, Lars is absolutely correct. The multiplier that I gave, of
    7819, is wrong, and quite significantly so. Lars' figure, of 7915.7 is
    indeed correct, for a spherical Earth. Sorry about that.
    Where did my erroneous multiplier come from? I had scribbled it into
    the margin of an Admiraly Navigation Manual, perhaps 20 or 30 years
    ago, and took it for gospel, when I should have checked it out. A clue
    comes in the line scribbled below it, in which I had noted the log of
    that quantity as 3.8986, which is in fact the log of 7918. So the 7819
    was a transcription error for 7918, exactly as Lars hypothesised. And
    even that number isn't quite right for a spherical Earth. The figure
    Lars gave, at 7915.7 (or 7916 for short) is the number that should be
    Note that navigators' tables for meridian parts will differ, slightly,
    from numbers calculated in the way explained above. That's because
    they take account of the Earth's shape, either the WGS84 reference
    ellipsoid or earlier approximations to an ellipsoid, and to different
    definitions of the mile on that ellipsoid, as Lars has explained.
    I'm sad to have added unnecessary complication to an already complex
    topic, but pleased that there are keen-eyed readers around, alert to
    spot such boobs.
    contact George Huxtable at george@huxtable.u-net.com
    or at +44 1865 820222 (from UK, 01865 820222)
    or at 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.

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