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    Re: finding your latitude through double altitudes (and elapsed time).
    From: Joel Silverberg
    Date: 2007 Apr 08, 14:30 -0700

    George, thank you for your comments in response to my inquiry.  In
    particular your description of the diagram with its auxilliary
    constructions was very helpful.  Your attempt to associate my
    variables with the diagrams was also helpful in helping me see the
    link between the diagram, the constructions and the equations.
    Frank's suggestion that I look at Benjamin Peirce's treatise provided
    me with just what I needed to make sense of all of these sources.   I
    have written up a detailed summary of the method, including a complete
    illustrated derivation of all equations and will email it to Frank to
    see if he wishes to post it to the navlist.  If you send me your
    email, I will mail you a copy as well.  (It is an 8 page pdf file).
    Cotter does not name any angles or sides with variable names, only
    points on the sphere.  That is part of what makes his explanation so
    difficult to follow.   I have retained the variables that Bowditch and
    Peirce use for sides and angles, while changing the names of points on
    the sphere to match those that you use in your posting and that Cotter
    uses in his text.  You are correct that my variable A   is half the
    great circle distance between X and Y as you note below.   Also, B is
    latitude (or declination)  of point M.
    Variable C is not MR as you suggest, but the distance from the zenith
    to R.  Variable Z (not to be confused with the zenith) is not the
    distance from the zenith to R, but the distance from M to R.  Variable
    E is not PR, but it's complement... that is E is the declination (or
    latitude) of point R.
    The relationship between E, Z, and B with its complicated sign rules
    is now easier to understand.  If E is the declination of R and B is
    the declination of M, then ... if R lies between M and P, then my
    variable z = declination of R - declination of M,  that is z = E - B,
    which solving for E gives E = B + z.   On the other hand, if M lies
    between R and P, then z = declination of M - declination of R .  That
    is z = B - E.  Solving for E, we have E = B - z.    (The little z in
    these equations does not refer to the zenith, but the the distance
    between M and R.  )  This clarifies Bowditch's explanation.   R lies
    between M and P when the zenith and the pole lie on the same side of
    arc XY, while M lies between R and P when the zenith and pole lie on
    opposite sides of arc XY.
    > The double altitude problem is also discussed at some length in Charles H 
    Cotter's "A History of Nautical Astronomy", 1968, and Joel
    > might find it useful to refer to pages 152-155. I have a serious distrust of 
    Cotter when he gets into anything mathematical, and a
    > long list of perceived errors in that book, collected by Jan Kalivoda and 
    me, can be found atwww.huxtable.u-net.com/cotter01.htm .
    > However, he has delved into many obscure texts, and provided information 
    which exists nowhere else. It's in interpreting them that
    > he seems to get out of his depth. It's a really useful book to have, but now 
    hard to find and expensive.
    > He looks into Ivory's method of 1821, and Ainsley's of 1867, both of which 
    seem to correspond with the Bowditch procedure that Joel
    > describes so well. His diagram, figure 7, corresponds exactly in its 
    lettering with Joel's descripion, but to understand what's
    > going on, some more construction is called for.
    > The initial presumption is that changes in the Sun's declination during the 
    observation are small enought to be neglected, as long
    > as a mid-value is used for declination.
    > First, Joel needs to join points X and Y by a great circle (not by the path 
    of the Sun, which is a great circle only at the dates of
    > the equinoxes). Because X and Y have the same declination, that great circle 
    arc XY is symmetrical about its vertex, the mid point
    > where it comes closest to the pole P. That mid-point is to be marked M. The 
    resulting two triangles PXM and PYM are both
    > right-angled (at M). The arc MX = MY is what Joel has taken as A. The 
    latitude of M then corresponds to B. And his first two
    > equations then follow from the rules for right-angled spherical triangles.
    > Next, some more construction. Join M by an arc to the pole P. Where that arc 
    passes closest to the observer's zenith Z, mark another
    > point R, and join ZR. MRZ and PRZ are now two more right-angled triangles 
    (at R). As I see it, Joel's C now corresponds to the arc
    > MR, his Z to the arc ZR, and E to the arc PR.
    > Now, applying more spherical trig to the right-angled triangles ZMX, ZMY, 
    and ZMR, brings him to his third equation (for C) and from
    > that to his fourth equation (for Z). I haven't even tried to follow the 
    details of those manipulations, but they are all in Cotter.
    > Then, if that's correct, I get a bit puzzled about his fifth equation, which 
    he states to be E = Z +/- B, because as I see it, those
    > angles are not in the same plane. It seems more likely to me that the 
    intention is to derive the lat (or perhaps colat) of R from
    > the lat (or perhaps colat) of M, using the difference angle C. So I think 
    that fifth equation should be combining B and C to obtain
    > E, and suggest that Joel might take another look at it.
    > Now we know E (= PR) and Z (= ZR) in the right-angled triangle PZR, so we 
    can compute the third side PZ, which is the colat (and
    > from that the lat) of the observer. which was what was wanted.
    > Note that I have completely ignored any subtleties in the complex sign-rules 
    that Bowditch presents, in aiming just to give a
    > general picture of how that complex calculation was meant to work. Hope I've 
    got it right, but not completely confident... Anyway,
    > it may give Joel a start-off in the right direction.
    > George.
    > contact George Huxtable at geo...@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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