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    Re: Longitude by observation of the Sun, 19th Century
    From: Frank Reed
    Date: 2013 Jun 12, 08:31 -0700

    Jacques, you wrote:
    "The "longitude method" is very "classic" : in France, we used the Borda's formula for the calculation of apparent time."

    When you say "we used Borda's formula", do you mean that navigators were advised to use Borda's formula in navigation manuals? Or do you mean that you have examined logbooks or other primary source evidence on this? It's perhaps worth noting that the standard method of solving a time sight in British/American navigational culture was considered so generic, so basic, that no one's name was attached to it. It wasn't, let's say, "Maskelyne's method" for working a time sight. It was just "finding the true time".

    And you added:
    "The problem is the latitude : if your DR latitude is incorrect, the apparent time is also incorrect. So, you calculate the longitude only after the observation of meridian altitude : with the latitude of 12 hours and the dead reckoning, you can calculate the latitude for the moment where you take the observation of the morning."

    This issue is often "over-sold". First, this is not some unique property of 19th century methodology. In modern terms, we are talking about a running fix. Many modern ocean-going yachtsmen have used daytime fixes from Sun LOPs as their standard celestial method. They take a Sun sight in the morning, then shoot another a few hours later. The first LOP gets advanced and crossed with the second. Of course, there is some uncertainty in this process since the sights are not simultaneous. This is nearly identical to the 19th century process, but expressed in a somewhat different mathematical formalism. And we can easily understand the potential pitfalls of 19th century time sights by thinking about them as modern LOPs.

    Suppose one clear morning at sea I shoot a sight of the Sun when it is bearing almost exactly due east at 9am. Later, I shoot the Sun's meridian altitude. A navigator using modern tools would have an LOP running north-south for that sight. The noon sight would yield an LOP running east-west. The morning LOP would be advanced based on course and speed in the intervening hours, and a running fix would be plotted. Note that there is no uncertainty in the longitude of this fix from any error in latitude. Since the LOPs are perpendicular, shifting the Noon LOP does not affect the longitude of the fix. The only uncertainty in longitude results from the small uncertainty in the DR during the hours since the morning sight. These same conditions are true, and for exactly the same reasons, if we work the sights in the 19th century fashion. A time sight calculated when the Sun is due east or due west has no dependence on latitude (to first order).

    Suppose the following day it's mostly cloudy. I shoot the Sun at 10:30am when it bears southeast. Later I can't get a meridian altitude at noon, so I estimate my latitude based on the DR from the previous 24 hours. I draw a "DR" line of position on my plotting chart showing that estimated latitude. Then I cross that with my morning Sun line. Unlike the previous case, where the LOPs were perpendicular, my morning Sun sight LOP crosses the noon laltitude LOP at an angle of about 45°. So there is now uncertainty in my fix both in latitude and longitude. The estimated latitude translates directly into an equal uncertainty in longitude. And the same was true, and for the same reasons, with respect to the 19th century "time sight" methodology. If the time sight was taken when the Sun was 45° from the meridian, then an uncertainty in latitude of some nautical miles translated into an equal uncertainty in longitude (in nautical miles, not minute of longitude).

    And Jacques, you wrote:
    "Another method is based on the "Pagel coefficient" (after 1847) ; this coefficient is used for "corrected" the longitude of the morning calculated with the DR latitude. But the correction is only possile after the observation of the meridian altitude."

    A little note of 'caution' here. There are all sorts of "methods" that made it into the textbooks that were never or only very rarely, in fact, used by practical navigators. Just a reminder that there is a difference between the history of navigational mathematics and the history of navigation. The history of navigation consists of the tools and methods navigators actually used in practice, as evidenced in their logbooks and their calculations, as opposed to what they might have been taught in a classroom or a textbook.

    -FER


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