NavList:

George, you wrote:
"Frank's discovery that short-cut methods are possible, simplifying the clearance of lunars, over quite a range of possible geometries. I think he has it right. It's highly inventive. But it isn't any sort of replacement for the real thing."

We're simply talking here about an alternative method of CLEARING lunars. Many such were invented in they heyday of lunars. They were ALL replacements for the "real thing," as you put it. The very efficient method of clearing lunars that I have described was apparently unknown in those days, and that's a darn shame since it would have saved an awful lot of time and it might have allowed a few more navigators to try out lunars. Lunars still suffered from all of their intrinsic limitations, but reducing the time wasted would have helped no matter what.

I provided one barebones test for using the clearing method that I've described. There are simpler tests. One could easily prepare a quick look-up table occupying no more than a double-sided octavo page. You make your observation... you check against the table... if it's in bounds, you can do the quick solution... otherwise, dig out the logarithms.

You wrote:
"But now, consider the situation facing the navigator who has never learned the full trig procedure for clearing a lunar. What happens if the test fails? He has no alternative but to pack up, and wait for another day."

But that's your imagination leading you astray. You're imagining a navigator who is lazy and has learned ONLY the easy method. But speculate again on a navigator who has learned both. That navigator need not waste an extra ten minutes on the logarithmic solution when the circumstances would permit an "easiest lunar" as they often do. Many nautical astronomers and mathematicians worked to reduce the computational burden of the lunarian navigator with all manner of tricks. Yet this one, which would have saved nearly all of the logarithmic work, was unknown.

And you wrote:
"But away from the tropics, the opposite-azimuth situation will arise only once
a month."

You're still picturing a perfect or perhaps "slightly out-of-line" vertical circle case. But that's not it at all. A WIDE number of cases can be converted into vertical circle equivalents because of the weak dependence of the lunar clearing process on the exact altitude of the Moon in many cases. These cases are more common in low latitudes, but they are not exclusive to low latitudes by any means.

And you wrote:
"That doesn't matter, to a hobbyist, who just wishes to test his skill when the occasion arises. He could go home and try again another day."

Let's be clear here: this has almost nothing to do with modern lunarian hobbyists who would most probably clear lunars by calculator or computer. For practical solution today, only those very few who wanted to see a solution on paper AND didn't enjoy spherical trigonometry would find an interest in this today.

And you wrote:
"But Frank was imagining himself back in the 1780s, when lunars were starting to be used in earnest by professional navigators at sea. They knew how to work them. They may not have understood what they were doing, but it was a familiar, well-memorised routine. [...] The full trig procedure had to be available, at his fingertips, for when it was needed."

So you're saying that they would have avoided any techniques that saved them time?! That strikes me as contrary to the history of the modern world itself, but maybe you're right (??). Maybe those 18th and 19th century navigators were just that inefficient, that addle-brained, that slow to learn. Well, no, we all know that they weren't... And yet EVEN IF the practical navigators were like that, why didn't this exceedingly easy method make it into the publications of the land-bound nautical astronomers and mathematicians? They printed nearly endless alternative methods of clearing lunars. But the catch is that their methods were, of course, aimed mostly at impressing other mathematicians. And there's the rub.

-FER

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