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I am having difficulty figuring out the logic behind the calculations
for finding the latitude using the method of double altitudes.  I am
working from an 1833 edition of Bowditch.   I do not know how far back
this particular method goes ... I
am sure it is earlier than 1833.  This edition of Bowditch gives three
methods.  The first method is the one I am asking about.  The second
method is identified as basically the method of Douwes and the third
method takes into account the possibility that the declination or
polar distance of the object at the second observation may differ from
that of the first observation.

I'll summarize what the method does, and how the algorithm proceeds
and then I'll ask my question(s).

The method is as follows:   determine the corrected altitude of the
sun at two points in time, noting the elapsed time between the two
observations.   You do not need to know the actual time of either
observation, simply how much time has passed between observation #1
and observation #2.  A series of logarithmic calculations lead
eventually to your latitude.

The method in its simplest form assumes that the ship is stationary
during the interval, but it is easily modified to take into account
movement of the ship between the observations using a dead reckonning
of the change in position between them.

The details, as described in Bowditch are as follows:

The sum of the log-csc of the elapsed time and the log-sec of the
declination is the log-csc of  arc or angle A.
(The values of numerical examples in Bowditch indicate that he is not
really using the the log-csc of the elapsed time as he claims, but the
log-csc of half the elapsed time).      The sum of the log-csc of the
declination and the log-cos of A determines the log-csc of B.    The
sum of the log-cos of half the sum of the two altitudes and the log-
cos of half the difference in the two altitudes is the log-sin of
C.    The sum of the log-cos of A, the log-csc of the half sum of the
alititudes, the log-sec of half the difference in altitudes, and the
log-cos of C determines the log-sec of azimuth angle Z.

Z is named north if the zenith and the north pole lie on the same side
of the great circle between the two observed  points  and is named
south if the zenith and north pole lie on opposite sides of that great
circle.  B has the same name as the declination.   E is the sum of B
and Z if B and Z have like names and is the differnce if B and Z have
contrary names.   Finally the sum of the log-cos of C and the log-sine
of E is the log-sin of the latitude.

I am struggling to see what this elaborate rule does in terms of the
spherical triangle   P-X-Y , the spherical triangle Z-X-Y, and the arc
P-Z  (where X and Y are the positions of the sun at the time of the
two observations, P is the north celestial pole, and Z is the
zenith.    (P-Z is the complement of the Latitude, P-X =P-Y =
complement of the declination,  Z-X is the complement of the first
altitude, Z-Y the complement of the second altitude, and X-Y is the
arc of the great circle between the two solar positions.

With the availability of computers and calculators, it is no longer
essential to view things in their logarithmic form.
Direct solution of the trigonometric problem is now easy and it is far
easier to see what the method is actually doing from a geometric or
trigonometric viewpoint without the logs in the way.    I assume that
the calculations involving half the sum of the altitudes and half the
difference in the altitudes derive from the trigonometric identity for
converting sums and differences of trig functions to products of trig
functions (so that logarithms may be used).

If I reexpress Bowditch's method without using logarithms or the sum/
difference to product identity, the calculations are equivalent to
solving the  set of equations below.

Let PD (polar distance) be the complement of the solar declination and
let theta be the hour angle corresponding to the elapsed time between
observations.  Let H1 be the altitude  of the first solar observation
and let H2 be the altitude of the second solar observation.

The rule is then captured by solving the following six equations:

    sin(A) = sin(theta/2) * sin(PD)         [solve this equation for
A]

    cos(PD) = sin(B) * cos(A)                 [solve this equation for
B]

    2 sin(C) * sin(A) = sin (H2) - sin(H1)   [solve this equation for
C]

    2 cos(A) * cos(C) * cos(Z) = sin(H1) + sin(H2)  [solve this
equation for Z]

   E = Z + B      (or E = Z - B as per rule)     [solve this equation
for E]

    sin(Latitude) = sin(E) * cos(C)            [solve this equation
for Latitude].

Can anyone help me understand what these angles and arcs represent.
Where in the diagram are A, B, C, Z, and E ??   How do these
calculations derive from the geometry of the situation ?   It seems to
me that A is the value of half the length of the great circle segment
connecting the geographic position of the sun and the time of the two
observations.  Why he wants half of the segment I do not understand.
I have been staring at these equations for weeks and can not see what
B, C, Z , and E are.     If there are auxilliary lines that have been
added to the original triangles to convert the oblique spherical
triangles to right triangles, then what are those lines, which
triangles are being considered and how to A,B,C,Z, and E relate to
these auxilliary figures?   If anyone can untangle even some parts of
this rule, perhaps it will put me back on track for figuring out the
rest.

I have seen a direct method explained in several sources, where the
declination and elapsed time are used to compute the length of XY,
then the law of sines is used to calculate one of the base angles in
PXY.  Now the three sides of ZXY are known and the law of cosines is
used to find the angles.  Subtracting the base angles gives enough
information  to find an angle in PZY  (or PZX)  and PY and ZY being
known, the side PZ can be found by the law of cosines and the
complement of PZ is the latitude.   But I have not been able to see a
connection between Bowditch rule and this direct method.   I don't
know if that is because there is no connection, but rather a totally
different way of solving the problem, or if I'm just not seeing it.

I hope someone can shed some light.... this is driving me crazy.


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