NavList:

As I mentioned Sunday evening, I decided to break my reply on this in half so that I could get the first part out on a timely basis. The first part is in post 11863 located here: http://fer3.com/x.aspx/11863.
And now here's the second half. It's long-winded, I know...

George H, you wrote:
"Navigators of that era knew what they were doing."

I wouldn't say that, or at least I would qualify those specific words. The vast majority of navigators throughout history have not really known what they were doing in the sense of having a deep, mathematical understanding of the subject. That wasn't their job. They learned rules and techniques which others assured them were the best for the task of navigation. The task of determining those rules and techniques historically fell to the community of "nautical astronomers" and, down-stream from them, the community of authors of navigation manuals. Unfortunately, nautical astronomers and to a lesser extent the publishers of navigation manuals were often disconnected from the realities of practical navigation. This has been a conflict many times in the history of navigation. It was not navigators at sea who chose not to do longitude by lunar altitudes. They had no choice. It was the astronomers and the textbook authors up-stream who made that choice.

You wrote:
"Their accepted way of doing the job squeezed all the possible accuracy out of it"

That simply isn't true, George. The historical methods, the ones that were ACTUALLY used in practice, frequently ditched minor details --minor details which were well-known and of great concern to the land-based nautical astronomers. To name just the simplest example, real lunarian navigators never corrected for the oblateness of the Earth (in all the cases in logbooks that I've ever seen). It was a minor detail. In fact, relaxing the accuracy of lunar distance calculations until they were just good enough for the purpose at hand was big business. Whether navigators knew how much accuracy they were losing is hard to say. Like today's navigators, I imagine most based their choices of lunar tables and methods on personal, anecdotal recommendations.

You wrote:
"And lunar-distance could be applied most, though not all, of the time; as long as the Moon, with Sun or star, could be seen in the sky"

In pure theory, yes. The land-based nautical astronomers, specifically Maskelyne himself, designed it with that goal in mind. But that's not how it worked out in reality. In the real world of actual navigation history, lunars were limited to relatively specific periods of time and generally were taken using the Sun only. Sure, you could use some specific stars, the nine (briefly ten) lunars stars, but this wasn't popular. This limited frequency of the practical application of the lunar distance method makes it much more comparable to "longitude by lunar altitudes" than theory alone would suggest.

You added:
"not just occasionally in special circumstances and from special places."

The special circumstances that applied to historical lunar distance observations were not all that different from the special circumstances that would apply to longitude by lunar altitudes. In any case, no one is suggesting pretending that longitude by lunar distances didn't exist. This is entirely a creation of your own imagination. But longitude by lunar altitudes was IGNORED as a viable method.

George, you wrote:
"Pretending that it was done otherwise, to minimise its calculational problems, is lazy teaching, short-changing any serious student
who seeks to know how it was done. It's an attempt to rewrite the history."

Now George, this is just plain DUMB. "Lazy teaching"? Re-writing history?? WHO are you suggesting is re-writing history? WHO are you accusing of lazy teaching? Do you know somebody who has done such things? Who are these people?? Are you suggesting that John Letcher was guilty of "lazy teaching" or "re-writing history"?

And you wrote:
"Frank seems to be aiming towards teaching celestial-navigation without trig."

You bet. If you can find a way to solve a problem that is easier, takes less time, and is less prone to error, then that's the way you should solve it. The entire history of modern nautical astronomy has consisted of inventing methods that minimize the mathematical grind. Just consider the most widely used tables of the late 20th century, namely H.O. 229. Their whole purpose was to relieve the navigator of the drudgery and the error-prone calculations of spherical trig by providing the solutions to the spherical trig problem in tabular form.

And you added:
"And that's not possible. If the trig is stripped out of it, it isn't celestial navigation any more; it's some sort of "pretend" game instead."

That is just plain nonsense. There are plenty of cases in navigation where problems in spherical trig reduce to addition. The "collapsed triangle" of the meridian sight, e.g. latitude by Noon Sun, is the most blatant example. While one might solve the problem of the meridian sight by plugging meridian values into the spherical triangle as a classroom exercise or a textbook lesson, it would never be done in practice.

Of course, I realize that many people who become celestial navigation enthusiasts today get into very much because they're looking for some fun math to play with. That's what got me thinking about the educational backgrounds of NavList members. We are a group rather top-heavy with physics-folks, you and me included. People who have studied math and physics professionally get some real pleasure out of the various ways of working spherical trigonometry problems and positional astronomy and all the rest. So maybe I was spoiling the party by discovering that somewhere around 50% of all practical lunar distance problems can be cleared without using spherical trig at all. But that's the way it is. This is not "pretend", George. It's mathematics. And sometimes seemingly difficult problems in mathematics can be mapped onto much simpler solutions. Who would chase a difficult solution when an easier solution is available?

Speaking more generally for a moment, consider an analogy from the game of pool (billiards, that is). I am sure that 90% or more of NavList members have played pool more than once. Like celestial navigation, it tends to be popular with people who know some math and physics because there's math and physics in it. As a player progresses from beginner to intermediate level, he or she learns how to make progressively more difficult shots. At first, the only shots that go in are the ones with short distances between the target and the cue ball and cases where the alignment of cue, target, and pocket are nearly linear. Then comes the realization that the angle at which the cue ball strikes the target ball determines the future trajectory of the target quite precisely and suddenly a whole slew of more difficult shots become possible. Then the player begins to learn how to bounce balls off the rails and realizes that this is just like bouncing rays of light off a mirror. This intermediately-skilled player then faces a choice: go after increasingly complicated shots always seeking difficulty because that's the source of pleasure in the game (harder shots are better shots, goes the logic) or step back and watch the masters at work... When an intermediate-level pool player observes a real master of the game at work, they almost invariably say that the master player always seems to get "lucky shots". They're not spending their time chasing down exceedingly difficult shots. Instead they have reached a level where they know that it's all about working the table so that every shot is CONVERTED into an easy shot, and really difficult shots are un-necessary (it's all about the leave, as they say). And in some ways, this is a complete change in the way you have to think about playing the game. If the point is to win the game, then you have to follow the lead of the master players. Rather than concentrating on learning ever more difficult puzzles, ever more difficult shots, you have to change your approach completely and learn how this process of setting up every shot so that the next shot is always easy. Now back to navigation...

If the goal of celestial navigation is to "win the game," meaning that the WHOLE POINT is to get a fix accurately and reliably, then there's no reason to fuss over chasing the most mathematically abstruse solutions to problems. If a problem can be worked by simple addition, and worked reliably and quickly that way, then that is the way to go. The goal is to convert the most difficult shots into easy shots. If there's a way to make celestial navigation "easy," then that's what a master navigator will do. On the other hand, if the goal of celestial navigation is to have fun with challenging math problems in a pleasant practical setting, then by all means, one should seek the most mathematically convoluted approaches to problems.

Regarding longitude by lunar altitudes, you wrote:
"A method that works only within a narrow range of low latitudes, close-azimuths, similar altitudes, and near twilight, is hardly a navigational technique at all. For use by navigators, it's wanted when it's needed, not when all the omens are right. Already, any celestial navigation method is limited enough, requiring clear skies. Any lunar technique is further limited, being ruled out near New Moon. Further restrictions are unwanted."

Unwanted by you. Fine. And of course you're not alone. Plenty of people who have expertise in celestial navigation get pleasure from the elitism of it. The very concept of an "easy" solution to a problem in celestial navigation is unpleasant to such people, maybe even degrading to the "dignity" of noble nautical astronomy. Who would want to take this beautiful elite subject, the preserve of those endangered creatures who have studied spherical trigonometry, and make it more accessible? Well, that would be me. And plenty of people like me.

You re-quoted your post from January 5, as follows:
"The deficiencies in the method, resurrected by Frank, are all concerned with the problems of using the horizon for precise measurement. Those problems are what the traditional measurement of lunar distance avoids. Being an angle between two bodies, up in the sky, the horizon plays no part in lunar distance (except in an auxiliary measurement which calls for no great accuracy). So a precise observation can be made, which depends only on the skill of an observer and the precision of his sextant."

George, you might as well go back to your posts from March of 2002 (over a year before I joined NavList). You're saying exactly the same thing. But you have "closed the book" on this subject prematurely. And I should reiterate that I largely agreed with you on this UNTIL I actually tried some of these longitude by lunar altitude sights and when I began watching the skies for those times when conditions were right --surprisingly often. When you try them out in the real world, your brain starts working again. You realize things that you didn't see in pure theory.

Here are some statistical comparisons for you to ponder:
1) what is the typical standard deviation of errors in lunar distance observations given some specific combination of observer and instrument?
2) if the observer in (1) using the same instrument observes altitudes of the Moon or a star above the twilight horizon, what is the standard deviation of the errors that you would expect in individual altitudes? (obviously larger, for all of the reasons you have described)
3) what is the typical standard deviation of the errors in DIFFERENCES in altitudes between a star and the Moon observed within thirty degrees of azimuth of each other at nearly the same time?

Case 3 is the CRITICAL case for longitude by lunar altitudes. It's much smaller than the standard deviation for individual altitudes in case 2. You are completely correct in pointing out that the horizon has uncertainties which affect individual altitude observations. Those uncertainties have a standard deviation on the order of 1 to 2 minutes of arc, and if that was the end of the story, it would be devastating for longitude by lunar altitudes. But it's not the end of the story (open that book again). The standard deviation for differences in altitude is quite a bit smaller. Let experiment decide it for you. Maybe it's not quite as good as case 1 (true lunar distances) but it's close, and it's good enough to illustrate the sorts of accuracies that navigators used to get with real lunar distance observations 200 years ago.

George, you wrote:
"Initially, the only weakness Frank recognised [...]"
Followed by:
"Now, he has come to recognise [...]"

What are you talking about here? I do apologize if I ever confused you or anyone else by not emphasizing that the observation of the second body's altitude should be nearly simultaneous and more or less in the same azimuth. You yourself deserve credit for emphasizing what could go wrong if the observations are not contrained in this way back in 2002. You made the point very well, but you didn't consider the simple solution: observe the objects above the same horizon.

You wrote:
"...line between the Moon's horns- "within 45 degrees of horizontal is good, within 30 is excellent". Hardly "good", if both bodies are shifting at 45º, which on its own will double the error compared to a conventional lunar distance."

What do you mean by "shifting at 45º"? IF the Moon's horns are tilted less than 45 degrees with respect to the horizon, then the Moon's motion on the celestial sphere, projected on the vertical, will be within 30% of its actual value at that moment of time. Unless I've missed something, the error would NOT be double. It would be increased by 30%.

And you wrote:
"But he neglects all the other problems that bedevil the horizon"

Nope. You're wrong here. The uncertainty in the horizon is either systematic for all sights, in which case shooting in more or less the same azimuth at more or less the same time eliminiates the error, or it's residual random error, which is MUCH smaller and also "bedevils" true lunar distance observations. So just how large is that residual random error for longitude by lunar altitude observations? And just how large is it for longitude by lunar distance observations?

And you wrote:
"It is, indeed, true that a good knowledge of index error is needed to measure a lunar distance, and also true that where two bodies are observed at nearly-similar azimuths, any effect of index error will cancel. That's the one-and-only aspect in which the lunar altitude method, used in that way, can possibly be described as "better"."

FIRST, this is NOT the only aspect of "longitude by lunar altitudes" that can be better than longitude by lunar distances. Above all, the aspect that makes longitude by lunar altitudes "better" is the very thing that Letcher and Chichester and the others brought up when first describing it. Longitude by lunar altitudes is a method that employs the navigators standard set of tools. In the mid-20th century and later, that means pairs of LOPs. Before then, it meant common time sights. The advantages of these sights are both observational (it is physically easier to hold a sextant vertically and shoot altitudes in the usual fashion than it is to shoot lunars at the awkward angles sometimes required) and calculational (navigators get lots of practice in common LOP calculations).

And you concluded:
"But is there any problem of uncertainty in the knowledge of index error? Not ever! If a navigator is uncertain about index error, its simply because he hasn't bothered to check it; the work of a moment. So that claim, that the method by altitudes can be "better" for that reason, is a spurious one. "

Huh. Interesting... I could have sworn that there have been a dozen or more discussions on NavList trying to figure out the best way to measure index correction and some real worrying over the fact that different methods produced results which differed by up to half a minute of arc. I can recall Alex Eremenko being very concerned over different index correction observations. And I seem to remember Bill Morris finding that he could get different values for index correction on the SAME instrument just by changing the telescope. And those are just two examples off the top of my head... Of course, a fairly good measurement of index correction, to the nearest half a minute or arc or so, is easy and only takes a "moment," just as you say. But testing for index correction at the level of accuracy required for lunar distance observations is really quite tricky. It's an essential component of lunars that is often over-looked. George, how many methods for checking index correction have you yourself tried? Do you know of a method that will reliably give you your index correction to a tenth of a minute of arc? What sort or reliability and repeatability do you get?? I actually have recently found a method that does give the index correction to the tenth of a minute of arc, again and again, reliably, and with near perfect repeatability. But it's a land-based method. So a navigator at sea, whether really trying to determine GMT because he has suffered that once-in-a-blue-moon failure and actually lost GMT itself, or if he is just engaging in a challenging experiment, might very well have reason to doubt a sextant's exact index correction. And in that case, yes, there is an actual practical advantage to longitude by lunar altitudes. Your "Not ever!" just isn't true.

-FER

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