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    Re: Measuring Dip in the 18th Century
    From: Frank Reed
    Date: 2013 Dec 28, 16:32 -0800

    For the potential accuracy of backstaffs and other pre-double-reflecting instruments, I suggested 10 minutes of arc. I want to clarify that this is being generous. Under excellent conditions with a fine instrument and top-notch observer, that's the best that you should expect. A reasonable expectation for normal conditions and normal observers and instruments would be 20-30' (in the 1 s.d. sense).

    Brad, you wrote:
    "I do agree with Alex's assertion, that because the resolution was to an arc minute, and that the early dip tables were to an arc minute, we can expect that they desired to observe to that resolution."

    Desire is a tough thing to quantify. But certainly some navigators knew no better and expected that they could measure the angle perfectly --never mind to the nearest minute! In fact, the concepts of statistical uncertainty and observational scatter were still centuries in the future. They measured and some worked their calculations to minutes of arc for one simple reason: it's next in order after degrees. In fact, we can find many cases from earlier centuries of positions and other quantities calculated to degrees, minutes, seconds, thirds, and fourths (yes, that's why "seconds" are called seconds: they're the second 60th-part division of angle, and thirds and fourths are similar 60th-part divisions after that). They worked these calculations to crazy precision because they knew no better. Of course, both you and Alex are correct that this means that some of those early navigators would have found merit in dip tables correcting for small numbers of minutes of arc. But this was blindness on their part. They were correcting noise. The question is whether these blind calculators represented the majority of practicing navigators or only a peculiar, delusional minority.

    While I'm here, consider a modern parallel. Imagine some "historian" in 200 years investigating early 21st century conceptions of the accuracy of GPS. This future historian would have decades of Internet data to pore over. Seconds after imagining the project, the statistics of billions of references to GPS positions would be available, and this historian would notice that many latitudes and longitudes are provided to one ten-millionth of a degree in our quaint era. Many calculations were performed on those extra digits. Would this historian conclude that we believed all those extra digits that were the default output from some of our devices and Internet services?? Perhaps this future historian would even deliver a PowerPoint presentation proclaiming the surprisingly high accuracy of our early GPS (only to be scolded by his or her third grade teacher for confusing accuracy with precision).

    And Brad, you wrote:
    "Those navigators had dip tables, back staffs and declination tables to an arc minute and USED THEM THUSLY."

    No, I don't agree with that --not yet :). We need evidence from primary sources, and textbooks are not primary sources, for the most part. The presence of a table in a textbook does not imply practice. It tells us what the author believed might be or should be useful to practicing navigators. On the other hand, the presence of that table in nearly EVERY textbook with many variants in style designed to make them "easier to use", could possibly be counted as evidence of wide use. But until you can find actual calculations on paper with such careful work, you've got a hypothesis without evidence.

    You also wrote:
    "They could not compare their results with double reflecting instruments. They could not know that they were only good to 10 arc minutes because they had nothing to compare it to. (The same can be said of celestial navigation just before GPS. Nothing to compare it to."

    I don't buy that! I agree with you that it is much easier to estimate the errors of celestial today, dramatically easier since it has been almost completely replaced by GPS. And that's a very good point. But long before GPS, even when the only reliable fixes were celestial fixes, it was still possible to make careful mathematical analyses of the accuracy of the method of celestial navigation, and it was also possible to make ballpark estimates from common practice with only minimal mathematical analysis. And of course we all know how this is done: you compare your work against the work of the guy standing right next to you. And you compare your work today against your work yesterday from the same location. There is no rocket science here. These methods of ascertaining repeatable, reliable accuracy are so obvious and so easy that they don't even need a name.

    If navigators really believed that their positions were accurate to a minute of arc because their observations were taken to a minute of arc, then of course there would have been no debate in navigation at all. Ships would have entered the English Channel without a care in the world since the latitude would have been known so exactly, or so they would have believed. Instead, even Samuel Pepys (I think I'm remembering this correctly) commented on the fact that navigators had angst-ridden discussions over the latitude and even the identification of the various headlands when in sight of land while entering the Channel in the late 17th century because in fact they had great doubts about their vessels' positions.

    -FER

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