NavList:

Douglas, you wrote:
"the inherent variability seems to be around one minute of arc measuring no matter what"

You should be able to do considerably better than that --at least when the other body is the Sun, Jupiter, or some of the other planets. With those objects as the "other body" in lunars, I routinely get results with a standard deviation of about a quarter of a minute of arc in individual observations and about twice as good when sets of four are averaged. My results are as much as two times worse with (true) stars as the other body.

And you wrote:
"This means I have either reached my limits of measuring with the equipment used; or there are still systematic errors I am not aware of and eliminating."

The collimation issue which I mentioned in another post is a likely candidate for lingering systematic error. The typical resolution of the human eye, as we've discussed before, is around one minute of arc or slightly better in good conditions. Since you're using a telescope with a magnification of 7x, you can can resolve angles as small as about 0.14 minutes of arc (or slightly better). Of course, it's always possible that there is some other issue with the sextant. I had a sextant for a long time that had micrometer eccentricity which added a seemingly random error of +/-0.5 minutes of arc to every observation. All of these things can be tested.

And you wrote:
"The other thing I tested was George's hypothesis that a simple mathematical mean (average) of the results should be as good as my graphs. I found it appears he is right: with these results anyway. "

Yes, this is usually true. Nonetheless graphing can be useful just to catch a really bad point. If you adopt any procedure for throwing out bad points, it's important to set standards in advance and live by them. Otherwise, you'll end up throwing out good data that only "appears" to make a more attractive graph.

You wrote:
"The main conclusion I have come to after this set confirm to me as found in my first experiments"

It's always nice when our experiments confirm our pre-conceived notions. It's also worrisome...

You wrote:
"that lunar distances are possible alright"

Of course they were possible --and practical, too. There is abundant evidence for this in the mountains of logbooks which have been preserved by museums. I've posted on many of these over the years on NavList. Perhaps it's time for a review...

You added:
"ignoring completely the question of the profound difficulties in the calculations that sailors had in the past"

This is a popular myth, but it IS a myth. Even before the first Nautical Almanac was published, Maskelyne and others had already published short, easy methods for clearing lunars, and these continued to be refined throughout the period when lunars were widely used. Of course, BEFORE the Nautical Almanac began to be published (starting in 1767), a lunarian navigator had to work out the Moon's ephemeris data (in effect calculating the almanac data for each sight) and that process reportedly took hours. But AFTER the Nautical Almanac became available, the calculational work for lunars amounted to only about twenty minutes and by 1830, more efficient methods had reduced the calculating time to less than fifteen minutes, possibly less than ten minutes. As for the supposed mathematical difficulty, the type of calculation was nearly identical to a common time sight which was standard work for any navigator back then.

You wrote:
"but the system is inherently inaccurate (or more correctly: of strictly limited accuracy with practical measurement with a sextant) because of the slow rate of change of the Moon's position with the background of stars."

Ok, but it's very easy to over-state this. As I have noted, I can routinely get lunar distances accurate to nearly a tenth of a minute of arc (standard deviation) when sets of four are averaged. A tenth of a minute of arc error in the angle is generally equivalent to twelve seconds error in the resulting Greenwich Time. To navigators spoiled by modern clocks that might sound pretty bad, but stop for a moment and recall the implications for the vessel's position. A twelve second error in time yields an error of only three nautical miles at the equator (or 2.1 n.m. at 45 degrees latitude). Two or three miles is NOTHING for traditional marine navigation. Even if you triple or quadruple these errors, you're still going to get you very close to your destination after months at sea. Additionally, lunars have another important property: they never get worse. A chronometer can change its rate and develop small random errors that accumulate over time. A chronometer (corrected with its assumed rate) may be in error by only three seconds, let's say, after two weeks at sea, but it might be in error by sixty seconds after three months. But you could always take lunars, and, as long as you had an almanac for the year in question, you could get the absolute time to within some fixed accuracy even after years away from a known longitude.

And you wrote:
"for astronomers the slow rate of movement of the Moon is OK, where seconds of arc measurements can be achieved with land-based equipment (large telescopes with large divided circular scales) fairly easily, but is severely limiting for sea going navigators."

Then why did thousands of sea-going navigators use them? Of course, the answer is that they were very practical for many decades. Almost immediately after the publication of the Nautical Almanac in the late 1760s, navigators in the Royal Navy and on well-funded British merchant vessels began using them widely. Of course, chronometers were a better solution especially when more than one was carried, but chronometers were rare and extremely expensive until the end of the 18th and into the beginning of the 19th centuries. Very roughly, from about 1767 to about 1790, lunars were used in conjunction with dead reckoning for longitude. From 1790 to about 1820 (on British vessels) lunars were used with chronometers for longitude. After about 1820, they were rapidly disappearing and a rarity on British vessels. For American practice, add around twenty to thirty years to all of these dates. Even on American vessels, lunars had essentially vanished from practical navigation by 1850 (though they lingered painfully long in navigation education).

So why did navigators STOP using lunars after about 1820 on British vessels and about 1850 on American vessels? The simplest answer is the price and abundance of chronometers. When it became possible to carry multiple chronometers, they acted as checks on each other, and lunars were then redundant. In addition, as Lecky notes, voyages were becoming shorter and more direct in the era of steam. And finally, the globe had been mapped. Even if you lose Greenwich Time completely, if you could land at any known island or port, you can recover GMT from the known longitude, or, better yet, ask for the GMT from another vessel.

And:
"so as a method in the right hands one cannot argue against it being practical and useful to sailors. I am just amazed they did it so well - better than I can achieve on dry land."

And bear in mind, it wasn't just experts who used lunars. There is abundant evidence in logbooks for lunars. And sometimes you can even find fully worked examples from back then. The calculations were usually thrown out, but every once in a while you can find one on the back of a page at the end of a logbook or in a notebook of calculations preserved by chance.

And you wrote:
"But then they did not have chronometers, so there was no alternative. If you wanted longitude you used Lunars or did without. From a practical navigating point of view I suspect even a 'rough' longitude obtained by lunar distance was of immense value and would make the difference between sailing one way to run down a latitude or the other way which was perhaps thousands of miles further"

Exactly so. There's a great story written in the memoirs of Basil Hall (direct inspiration for Patrick O'Brian's novels) where he described being put in command of an American schooner (a "Baltimore Clipper") which had been taken under suspicion of running the blockade of Napoleonic Europe in 1807. He had to sail from just off the US east coast, generally toward the southwest, to Bermuda. Without a chronometer, he relied on dead reckoning for longitude but decided to shoot some lunars to double-check that longitude. He was in the latitude of Bermuda and expecting to sight it to the east after sunrise the next day. To his surprise, he found that his longitude was already east of Bermuda. He was so convinced that his dead reckoning was correct that he shot another set just too be sure and got the American captain to work up a set, too, since he wasn't entirely confident in his work (he was a young man at this time). Sure enough, they turned back to the west and found Bermuda. I've written this up in detail in an old NavList post...

Another excellent example can be found in the logbook of the ship Sabina in 1849. The captain was an old salt with decades of experience on whaling ships, but on this voyage he was on his way with dozens of other men to find his fortune in the gold fields of California. Just a couple of weeks after leaving New York in the mid-Atlantic, he decided to check his longitude by lunars and discovered that his lunars differed his chronometer by some two degrees. Even though this is very late in the era of lunars, he trusted them and expressed his dismay over the performance of the chronometer. I also wrote this one up in detail in an old Navlist post...

I'll have more to say about your lunar sight data in another post. Just for now, I would mention that you should not be using Vega for lunars if your goal is to get Greenwich Time. Vega is too far from the ecliptic. Since the Moon's motion along its orbit is nearly along the ecliptic, the rate of change of the lunar distance is slower for objects at high ecliptic latitude. In the case of your sights, this DOUBLES the error in the calculated Greenwich Time. There's nothing wrong with using stars outside the standard lunar set IF you compare your cleared angle with the predicted angle and leave it at that. But if you go on from there and interpolate for GMT, it's important to use objects that are nearly aligned with the Moon's motion through the heavens.

-FER

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