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    Re: Sumner's Line (Navigation question)
    From: Frank Reed CT
    Date: 2006 Feb 7, 01:45 EST

    Bill wrote:
    "An ID 10-T  question."
    First, thank you for making me look this up. To save anyone  else the effort,
    if you write out ID-10-T, and drop the dashes... and make the  "1" look more
    like an "I", you get IDIOT. Myself, I don't believe in "idiot"  questions.
    They're usually GOOD questions.
    And so Bill asks his idiotic  question :
    "I continually see references to the classic "time site,"  but none of my
    texts actually spell out how it was done or the equation(s)  used.  The best
    I can surmise is that with a know/given latitude and  time, longitude may be
    Any help on how it was practiced and  and the math behind it (list or on-line
    references) would be  appreciated."
    A very good query, Bill. A time sight is the 20th century  name for the
    classic sight used in navigation from the late 18th century through  the 1940s to
    determine local time at sea. It is, in many ways, a remarkably  simple thing.
    Taking a time sight turns your sextant into a sundial. When we use  a sundial,
    aligned to the correct latitude, we read off the Sun's local hour  angle in
    hours and call it the time. A sundial also has to be aligned exactly
    north-south, but we can live without that alignment if we have an almanac  available.
    From the almanac, we can get the Sun's declination (we still need the  correct
    latitude for a time sight, just as with a sundial), then the Sun's  altitude,
    which can be measured accurately with a sextant, automatically yields  the
    Sun's local hour angle. This hour angle is exactly the same thing as the
    observer's local apparent time. In other words, if I measure the Sun's altitude,  and
    do a little math on it (see below) and get, for example, 45 degrees, then  the
    local apparent time is just 3pm exactly (or 9am if the Sun is east of the
    meridian). To reiterate, a time sight with a sextant performs exactly the same
    task as a sundial yielding the same information, the local apparent time, much
     more accurately and without the need for an exact north-south  alignment.
    So what's the math that converts a measured altitude into an  LHA? This is an
    easy spherical triangle problem. The Sun's local hour angle is  the angle
    measured at the elevated celestial pole between the observer's  meridian and the
    arc from the pole to the Sun (draw it!). To calculate that  angle, we can use
    the three sides of the spherical triangle made by the Sun,  Zenith, and the
    Elevated Celestial Pole. I won't spoil the fun. You try it out  for yourself.
    Draw that triangle and solve for LHA using the cosine formula.  That's it! On a
    calculator, it's easy. Note that it was popular in the
    logarithms-paper-and-pencil era to solve this triangle using the haversine  formula instead of the
    cosine formula. It's a little longer, but you don't have  toworry about as many
    "cases". That's what you'll find in old editions of  Bowditch. It's
    conceptually the same thing as the cosine formula --just a  different way to skin a
    After we've done our time sight with our  sextant (or read our sundial), and
    we have learned that the local apparent time  is, say, 3pm exactly, what next?
    As we know, the difference between local time  and absolute time, typically
    GMT, is exactly the same thing as the difference in  longitude between the
    observer's location and the absolute time location,  typically Greenwich. But
    there's a catch. Local apparent time runs a little fast  or slow during the year
    compared with accurate clocks. That difference is the  so-called "equation of
    time". So if you use a sundial and you want to know if  it's accurate, you need
    to have a table of the equation of time handy. Many  sundials in public
    settings have tables or graphs affixed to them or posted  nearby (unfortunately,
    for most people, these tables usually create the  impression that the sundial is
    "broken"). For out time sight, if we're going to  determine longitude by
    getting GMT from a clock, then we need to correct the  time sight by adding in an
    equation of time correction taken from the almanac.  Before chronometers came
    along, there was an alternative approach. With lunar  distances, it was easy
    enough to tabulate predicted geocentric lunar distances  directly in "Greenwich
    apparent time", in effect making the celestial "lunar  chronometer" read
    apparent time instead of mean time. That way, if it's 3pm by a  local time sight
    and 9pm by a lunar distance sight, the longitude difference is  simply 6 hours
    (or 90 degrees) exactly. No correction required. Until 1834, the  tabulated
    lunar distances in the Nautical Almanac were listed for every three  hours of
    Greenwich Apparent Time. After that year, they're published for every  three
    hours of Greenwich Mean Time, for better comparison with chronometers.  There
    were also "novelty" chronometers designed to read apparent time instead of  mean
    time. This was really pointless, so they didn't last.
    Notice that a  time sight, like a well-adjusted sundial, requires accurate
    knowledge of  latitude. This dependence on latitude is minimized if the Sun
    bears nearly due  east or nearly due west. Alternatively, you can take a sight to
    determine  latitude simultaneously with the time sight, but this was rarely
    done in  practice until the 20th century. Time sights started to become obsolete
    with the  rise of the "New Navigation" (line of position navigation) in the
    late 19th  century, but they were still widely practiced as the standard sights
    for  determining local time, and hence longitude, as late as the 1940s.
    One  more time:
    Sun's measured altitude (corrected) ---> cosine formula --->  Local Apparent
    Local Apparent Time + Equation of Time ---> Local Mean  Time
    GMT (from chronometer or lunar) - Local Mean Time --->  Longitude
    42.0N 87.7W, or 41.4N  72.1W.

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