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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
consistency.

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.

-FER
www.HistoricalAtlas.com/lunars


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