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    Re: shortest twilight problem...
    From: Frank Reed
    Date: 2010 Jun 30, 20:19 -0700

    Robin, you wrote:
    "Using the cosine rule of spherical trigonometry it can be shown that
    0 = sin(d)*sin(L) + cos(d)*cos(L)*cos(LHA1)
    cos(90+18) = sin(d)*sin(L) + cos(d)*cos(L)*cos(LHA2)"

    Yes, this is the standard formulation of the problem (at least in "modern" terms). And with some trig gymnastics, see below, it can yield the solution given by Joel Silverberg. And this problem, incidentally, is a close cousin of the "Latitude by Double Altitudes" problem that also used to fascinate the mathematicians somewhat out of proportion to its practical significance. In the latter case, we want the latitude from a given difference pair of altitudes and observed difference in LHA (from a common watch), while in the twilight duration problem, we want the difference in LHA for a known latitude and a fixed pair of altitudes.

    There's a simple geometric trick for the minimum twilight problem with no calculus required. Go here:
    http://books.google.com/books?id=jM4MAAAAYAAJ&pg=PA170
    and read article 219.

    There are numerous examples of the rule for calculating shortest twilight, without derivation, with instructions for working the rule in logarithms. See for example Mackay's "Theory and Practice of Finding the Longitude" from 1810 (which is excellent on many topics from nautical astronomy, spherical trigonometry, and also the history of the longitude problem broadly). It's also in Kerigan in 1838, etc. These are "kitchen sink" sources so the presence of the rule for working the problem does not imply practical value.

    There are also various derivations of the equation itself available. Here's another:
    http://books.google.com/books?id=qZU-AAAAYAAJ&pg=PA253
    using a "minimum" of calculus. :)

    And here's a version of it by Laplace:
    http://books.google.com/books?id=TZAEAAAAYAAJ&pg=PA366

    If you want more, go to Google Books Advanced Search, located here:
    http://books.google.com/advanced_book_search
    and enter "shortest twilight" in the 'exact phrase' box.

    You may want to limit the upper date to 1899 for some searches like this, but in this case leaving it off turns up one interesting modern result. There's a book by Heinrich Doerrie published in 1965 entitled "100 great problems of elementary mathematics: their history and solution" and making the list at number 97, just three steps from the bottom rung, we find "The Problem of the Shortest Twilight". Doerrie writes, "On what day of the year is the twilight shortest at a place of given latitude? This problem was posed, but not solved, by the Portuguese Nunes in l542 in his book de crepusculis. Jacob Bernoulli and d'Alembert solved the problem by means of differential calculus..." This book is in preview only.

    This was clearly a "puzzle problem" historically with little or no practical value, and it says a lot that there are almost no references to it in 20th century or later works (except Doerrie's history and various reprintings of early works). Its interest to mathematicians and calculators had clearly come to an end.

    -FER

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