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    Re: shortest twilight problem...
    From: George Huxtable
    Date: 2010 Jun 30, 21:50 +0100

    Robin succinctly described the problem of date of shortest twilight-
    "The interest of this problem is presumably in its apparent simplicity,
    relative mathematical intractibility, and counterintuitive result."
    That was beautifully expressed. I wonder if his numerical solution can be
    reconciled with the analytic expression quoted by Joel, as follows-
    "The declination of the sun on the day of shortest twilight is equal to the
    product of the tangent of half the twilight angle (half of 18 degrees for
    astronomical twilight) and the sine of the observer's latitude with the
    caveat that the solar declination on that day will be negative if the
    observer is in the Northern hemisphere and positive if he/she is in the
    Southern hemisphere.
    sine(declination) = tangent (9 degrees) * sine (Latitude) (declination is N
    or S as described above)"
    It seems astonishing that mathematicians were already using calculus, in
    its very early days before 1700, to tackle such tricky problems; if only as
    an exercise for the student.
    contact George Huxtable, at  george@hux.me.uk
    or at +44 1865 820222 (from UK, 01865 820222)
    or at 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.

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