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    Re: Sextant calibration.
    From: Frank Reed CT
    Date: 2007 Apr 21, 22:23 -0700

    Hi George.
    I agree with you that it would be a bit of a challenge finding enough
    steeples to make this work, but then again, there was a steeple for
    every few hundred people back then and almost no taller buildings.
    That will add up pretty quickly. A bigger problem is finding a flat
    city since this method will only work if the distant markers are all
    nearly lying along a great circle. Let's drop for now the historical
    case and consider the modern case. So I'll stop calling them steeples
    and refer to the markers as lighthouses. There are places on the shore
    in Connecticut where you can see two dozen lighthouses or similar
    fixed distant markers. And there are enough peninsulas and small
    islands that you can be almost completely surrounded by water with
    bits of land at varying distances in every direction. Lots of rocks
    and shoals to bump into so lots of aids to navigation which are just
    about perfect for this project.
    In my earlier post, I gave an example where the four lighthouses were
    located at the cardinal points of the compass and almost exactly 90
    degrees from each other. I set it up that way just to keep the example
    simple. This 90 degree match is not a critical condition. Let's
    suppose I have four lighthouses in the following azimuths relative to
    my position A: 28, B: 110, C:201, D: 296. Note that I don't need to
    know these values, but it might be that I've measured them by eye with
    a handheld compass, or maybe I've taken them off a nautical chart. The
    key thing here is that the approximate relative angles between the
    lighthouses are AB: 82 degrees, BC: 91d, CD: 95d, DA: 92d. Now suppose
    I measure these angles with my sextant and add them up. If I get
    anything but 360 degrees then the result is the total error from the
    four measured angles. Let's call the remainder R. Let's call the arc
    error as a function of measured angle
    arcerr(x). From this set of measurements, I know that
       R = arcerr(82) + arcerr(91) + arcerr(92) + arcerr(95).
    Now, if the arcerr function is linear or at least reasonably close to
    linear, then R/4 will be the actual arc error at the average value of
    those measured angles which is, of course necessarily, 90 degrees.
    Note that it does not have to be true that the arc error is the same
    at each of those angles. It only has to be true that the arc error is
    smoothly changing (linearly) over that range. Just as an example,
    suppose the arc error at 80 degrees is 1.0', at 90 deg it's 1.5' and
    at 100 deg it's 2.0'. In that case, the actual errors at angles of 82,
    91, 92, and 95 would be 1.10, 1.55, 1.60, 1.75 respectively. And of
    course, if later measurements prove this assumption of linearity to be
    seriously incorrect, it is only a matter of simple accounting to go
    back and fix it at the end.
    This first step, using a set of objects spaced all around the horizon,
    is a bootstrap step. We need it because the only angle we can count on
    with any certainty is 360 degrees in a complete great circle. But
    after we're done with it, we no longer need the whole horizon since we
    have now nailed down one good value for actual arc error of our
    sextant. If we know that the arc error close to 90 degrees is 1.5
    minutes of arc, we can then use that knowledge on another set of
    distant lighthouses, quite possibly viewed from an entirely different
    location. These should be spaced more or less evenly over one quadrant
    (90 degrees) of the horizon. For example, let's suppose we have
    lighthouse azimuths as follows, W: 60, X: 85, Y: 115, Z: 147. Now we
    measure all six possible angle pairs: WX, WY, WZ, XY, XZ, YZ. We
    correct WZ using our known correction for 90 degrees from the
    bootstrap step. Then using the same logic and similar arithmetic, we
    can derive corrections for 30 degrees and 60 degrees. And so on...
    The critical factor in all of this that let's us measure arc error
    without a calibration standard is the fact that angles measured along
    a great circle arc must add up. And if we measure all the way around a
    full circle, they have to total 360 degrees. It's just internal
    You asked whether the "Tratado de Navegacion" by Mendoza y Rios has
    been translated. I'm not aware of a translation. I think the topic is
    a bit too obscure to support a complete translation effort. It's a big
    fat tome. I wouldn't be surprised if it makes its way, untranslated,
    onto google books one day soon, but it's a very rare book so it may be
    tough to find a library willing to subject it to the stress of
    scanning. I read a substantial portion of it at the library in Urbana-
    Champaign, Illinois back in the summer of 2004. Meanwhile, one other
    major navigation book by Mendoza y Rios is now available on google
    books. It's his "Tables for Facilitating the Calculations of Nautical
    Astronomy" published in 1801 (I also own a copy of this one). The
    dedication page reads, "To the Reverend Nevil Maskelyne, D.D. F.R.S.
    and Astronomer Royal, whose ingenious labors have greatly advanced the
    science of astronomy, and whose indefatigable exertions, both in his
    public capacity and private studies, have contributed in a signal
    manner to the improvement of navigation, this work is, with all
    deference, dedicated, by his most obliged friend, and obedient, humble
    servant, Joseph de Mendoz Rios". It's the usual dedicatory fluff you
    see in that period but it's still fun to see. Mendoza Rios was also
    counted as a close friend by Joseph Banks. And by the way, since this
    book dates from many years after his arrival in England, it is written
    in English. Of his English, Mendoza Rios writes, "There are
    inaccuracies of another sort, which I am sensible will be discovered
    in the language of my composition; but the candid English Reader will,
    I hope, readily excuse them in an Author who is a native of another
    country". He's being modest.
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