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Once one has cleared the distance, one then needs to determine what GMT
that distance corresponds to.

The Nautical Almanac from those days listed the lunar distances for the
moon and various objects at three hour intervals. The cleared distance
would lie within one of those intervals.

The basic idea is to use interpolation to find the time corresponding to
the cleared distance. If D is the difference between the tabulated
distances which bracket your cleared distance, and d is the difference
between the earlier tabulated distance and your cleared distance, and t
is the elapsed time between the earlier tabulated distance and your
cleared distance, then:

      D   3 hours
      - = -------
      d      t

or,
                 3 * d
      t(hours) = -----
                   D

or,                         d
      t (seconds) = 10800 * -
                            D


O.K., this is pretty straightforwards, but Nevil Maskelyne recognized
that for the sailors of the time, this calculation was rather tedious.
It would require four entries into log tables to solve.

Maskelyne came up with a way to simplify this calculation. He define the
"proportional logrithm" of time t as being the common log of 10800 minus
the common log of t (seconds). Maskelyne then created a table of
proportional logs (to four digits) and included them in his famous
"Tables Requisite".

Here is how you use them:


Because:

    D * t = 3 * d

we can write:

   log D + log t = log 3 + log d

or,

   log t = log 3 + log d - log D

or, (subtracting each term from log of 3 hours):

   (log 3 - log t) = (log 3 - log 3) + (log 3 - log d) - (log 3 - log D)

or,
    (log 3 - log t) = (log 3 - log d) - (log 3 - log D)

This is therefore:

   plog t = plog d - plog D

where plog is the proportional log as defined above.

This trick means that the sailor only has to make two entries into the
prop log table (because prop log D was given in the almanac).

Now, only one problem remains, over three hours the non-uniform motion
of the moon means that simple interpolation will not yield enough
accuracy. To correct this the "correction for second differences" was
applied.

If the values a,b,c, & d are the results of some varying function, then
the first differences are the difference between the values.
I.E. a - b, b - c, c - d, etc.

The second differences are the differences in the differences,

I.E. (a-b) - (b-c), (b-c) - (c-d), etc.

Note that:
      (a-b) - (b-c)

     = a - b - b + c

     = a - 2b + c

If the interpolation is carried to second differences, then you get a
result accurate enough for navigation. The Nautical Almanac used to
contain a list of differences in the prop logs for each interval of t.
I.E., the prop log for each D is given, and the mean difference in the
prop logs for each pair of D's. (that is the second difference value)
which is basically the velocity of the moon relative to the object at
that time. One of the neat features of this method of tabulating the
distances is not only does it shorten the calculations required by the
navigator, but the differences in prop logs (the second difference
numbers) give you an indication of how fast the moon is moving relative
to the tabulated objects. Obviously you want to pick the fastest rate of 
change to get the best lunar, so the table tells you which is the best
object to select!

Those guys were no dummies eh?!

                            ^
                            |
                            |
              (Certified genuine Canadianism)


Cheers!

Jeff.


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