# NavList:

## A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding

**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 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 --~--~---------~--~----~------------~-------~--~----~ To post to this group, send email to NavList@fer3.com To unsubscribe, send email to NavList-unsubscribe@fer3.com -~----------~----~----~----~------~----~------~--~---