### A2 - - - CALCULATED MERCATOR SAILING - - - 10/03/2004

At the equator a degree of longitude is equal to a degree of
latitude, and they are approximately equivalent to 60 nautical
miles but in receding from the equator and approaching the
pole, while the degrees of latitude remain always of the
same length (but for a slight change due to the fact that
the earth is not a perfect sphere), the degrees of
longitude become less and less, thus the degree of
longitude at 70 degrees North or South of the equator is about
20 nautical miles.
In the Mercator charts the degrees of longitude are made to appear the
same length over the whole chart. It becomes necessary in order to
preserve the direction between different locations to increase the
North-South distance on the chart and such increase must become
greater and greater the higher the latitude.

The length of the meridian as thus increased between the equator and any
given latitude (L), expressed in minutes of longitude at the equator
as a unit, constitutes the number of **meridional parts (M)**
corresponding to that latitude.

The value for M is normally taken from a table of meridional parts, but
now that we are in the computer age, the value for M can be calculated
by a programmable calculator or computer. For normal navigational
purposes the value of M is taken to one decimal place (NNNN.N), but
for survey work greater precision may be required.

**Meridional parts **for the Clark spheroid of 1866.
M = 7915.704468 * log tan ( 45 + (L/2))
- 23.268932 * ( sin L )
- 0.052500 * ( sin L )^3
- 0.000213 * ( sin L )^5

When sailing from point 1 (L1, LO1) to point 2 (L2, LO2)
M1 = meridional distance for L1
M2 = meridional distance for L2
m = (M2 - M1) = (meridional difference)
C = course angle
DLO' = difference in longitude expressed in minutes
D = distance from point 1 to point 2
tan C = DLO' / m
D = 60 * ( L2 - L1 ) / cos C (nautical miles)

When L1 is greater than L2, then the angle C will be negative (fourth quadrant).
To put the answer into the second quadrant add 180 degrees.

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