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    Re: Cook's Incremental Reckoning
    From: Frank Reed
    Date: 2010 Mar 20, 20:17 -0700

    George H wrote:
    "Even as late as 1767, such an experienced navigator as James Cook was not
    noting his longitudes as from Greenwich, in his North Atlantic crossings.
    They were always from the point from which he took his "departure" ...Never from Greenwich, much to my surprise."

    First, I'm surprised George was surprised by this. There was no reason or even much value in expressing longitudes relative to Greenwich until after the Nautical Almanac and the lunar distance tables revolutionized the practice of longitude.

    Then Brad, you replied:
    "This means that he was using incremental reckoning, not absolute reckoning. Truly interesting! The problem with this type of reckoning is the accumulation of round-off."

    No. The choice of a starting longitude has nothing to do with any incremental issues or any round-off error. There's no problem picking some random meridian as the starting longitude for a voyage unless you need to look up navigational data, from an almanac e.g., that references a specific meridian. Choosing the Lizard, or Ushant, or Montauk as your meridian of departure does not make navigation less accurate. It does, however, make it more difficult in the long-term to compile and compare navigational information from different vessels and different voyages.

    Brad, you added:
    "Example: I went west 1.234 degrees on day 1 and on day two, I went west 2.345 degrees. Since that is 'too many' digits to write, on day 1, I record 1.2 degrees and on day 2, I record 2.3 degrees. How far west am I? According to my records 3.5 degrees, when I actually went west 3.579 degrees. Make this 1000 days, and you get the picture."

    Nope. Round-off error only becomes an issue when the navigation is MORE accurate than the precision of the calculations.

    You wrote:
    "Cook was out exploring for years, if I have it right."

    That was later. These trans-Atlantic voyages were on well-known routes and relatively short.

    And Brad, you wrote:
    "Therefore his records must show a gradual shift in longitudes with respect to time."

    Surely you must see that the uncertainty in the dead reckoning calculation would swamp any effect from the round-off error that you're worried about. Round-off adds a small "random walk" to the data. But if the step size in that random walk is well below the normal uncertainties in the process of DR navigation itself, then the round-off is irrelevant.

    And Brad, you wrote:
    "So he under-estimated by about 5% on each Atlantic crossing. Perfect example of precisely the effect."

    Whoa. Not a chance. This has nothing to do with round-off error. It's actual error in the process of dead reckoning. Some of it due to the poor instrumentation --a log line, an uncertain compass. Some of it due to the uncertain ocean --currents were largely unknown. It was considered normal practice to keep the reckoning *ahead* of the ship, even by intentionally making the log line the wrong length. The idea was that the calculation would show you in shallow water a bit before you actually got there. George seems to be saying that Cook's trans-Atlantic voyages had the reckoning *behind* the ship, which is unusual. This business of messing with the reckoning was enough of an issue that Maskelyne even devoted an appendix to the length of log lines in his 1763 British Mariner's Guide, which was fundamentally a work on longitude by lunars.

    -FER


    -FER

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