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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
integral
| 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
out
| 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
used.

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.

George.

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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