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George, you wrote:
"I suspect that the discrepancies between various calculations, that Andres
refers to, result from imprecise definitions of the nautical mile. If it's
assumed to be the length around theEarth's surface that subtends an angle of
1 minute between two local verticals, then when the Earth is taken as an
ellipsoid, that length varies (a bit) with latitude. It's even different, at
the same spot, when measured in different directions, because the Earth's
curvature is different measured North-South to what it is measured
East-West. For most purposes, we compromise on an average value of 1852
metres, which would be appropriate for a sphere. However, for geodetic
purposes of precise survey, a value for the "geodetic mile" is taken that
puts exactly 360 degrees, or 21600 miles, around the Earth's equatorial
circumference, making a geodetic mile 1855.3248 metres."

Yes. Quite so. The Bowditch version of the equation assumes that each degree 
of longitude at the equator contains exactly 60 nautical miles. I don't quite 
agree with Andres that the Bowditch formula is an "error in concept", though 
I see how he came to that conclusion. It's a scaling discrepancy only, and 
therefore a matter of units. Whether this is a literal error or not (even a 
tiny error) depends on what we do with the numbers next. If the numbers are 
then employed in a way that uses the same definition of a nautical mile, 
there is no problem. But it seems to me that this formula should have been 
updated to reflect the modern definition of a nautical mile. The difference 
is a small one regardless. 

According to Wikipedia (take it or leave it), the modern nautical mile was 
defined by by international agreement in 1929 and adopted by the U.S. on July 
1, 1954.

I quickly looked through some copies of Bowditch. This formula, with only one 
trivial change, has been in Bowditch since the major revision in 1881. From 
1837 to 1881 Bowditch's navigation manual was largely unchanged with only 
small insertions (like a paragraph on Sumner's method tucked away on a 
previously nearly empty page --so as to save on making new plates for the 
whole volume) and some deletions of obsolescent information. Through that 
whole period, meridional parts were defined, as Brad gave previously from his 
1849 edition, with an assumed spherical Earth. Starting in 1881, when the US 
Navy finally released its comprehensive overhaul of "Bowditch", the formula 
that Andres has quoted appears. This continues to the present day, and it is 
founded on the assumption that a nautical mile is exactly one minute of arc 
of longitude at the equator (no update with the heavily revised 1958 edition 
of Bowditch which would have been the first to reflect the new international 
nautical mile). The only very small change occurred a few years later when 
the exact value of the flattening was adjusted to reflect a slightly 
different spheroid model. 

-FER



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