NavList:

George, you wrote:
"And indeed, Janet Taylor has got it absolutely correct. Moore has set himself a quite impossible example. With the altitudes that he has given for the two bodies, there isn't nearly enough spacing across the sky, in any possible geometry, to achieve such a large value of lunar distance. It's a contrived example, that can't possibly happen in real life. Yet he has gone blindly through the motions of making the corrections to clear that absurd value, by two different methods, and indeed, got the same result in each case."

I never noticed Taylor's comments on this. But she's wrong. Well, no, not really wrong... she's just missed the point. The process of clearing lunars is rather robust mathematically, and observational error in the altitudes is no problem, up to a point --even if it seems to yield an impossible spherical triangle (though not as extreme as in this case). Also, inconsistent data don't really enter into the process of clearing as outlined here. It doesn't "hurt" the examples, though it certainly could confuse the student without some kind of comment. Even a footnote would help.

Taylor blamed Moore for the inconsistent example, and she added (copying from your post):
"I here observe, that according to 20th 12 Euclid, it is impossible and absurd that one leg of a spherical triangle should be greater than the other two".

Though Moore was long dead by the time she was writing these words, Moore already had an answer for her. In one my talks at Mystic in June, I spoke a bit about some of the feuds among authors and publishers of navigation manuals. They were, after all, vying not merely for public acclaim and bragging rights, but also for profit. And, of course, celestial navigation has long attracted pedantry and pomposity for a variety of reasons. We all have encountered people who know just enough navigation to consider themselves experts and geniuses when they still have a very long way to go. Moore knew this, and in the introduction to the "New Practical Navigator" from 1794 (or earlier?), Moore wrote:
"I am well aware that there are persons, who, to shew there own superior abilities in an obscure Club, will quibble and carp at some parts, and say, that they see nothing new, &c. To such Critics it may be answered, that a Triangle was a Triangle before the days of Euclid, and so it is now..."

So take that, Janet! And it wasn't his example anyway...

George, you wrote:
"Much of the impact of Taylor's point, made in 1833, is however lost, because that mistake occurred some 38 years earlier. By the time Moore's 18th edition of 1810 (also on Google) had appeared, that erroneous example had been dropped."

Not to worry. There was a vitriolic exchange of letters to the editor of the London Gazetteer back in the 1780s between an anonymous complainer who signed himself Nauticus and none other than Nevil Maskelyne. You see, it was Maskelyne, the Royal Astronomer and great lunarian himself, who originally published this inconsistent triangle in the Tables Requisite back in 1780. And he answered the complaints in some detail and with considerable anger, too. I've got a copy of this exchange of letters somewhere, but I'll have to dig it up. Moore, a decade later, was simply copying from the T. Reqs. By 1833 when Taylor was writing, it was already long after the heyday of lunars in Britain (and near the end in America) and the history was already being lost.

This anonymous Nauticus, by the way, also made the point that one could very easily clear lunars, at least in tropical seas when they were, in fact, most useful, by waiting for vertical circle circumstances. When the Moon and the other body are aligned in a vertical circle, either both in exactly the same azimuth or both on exactly opposite azimuths, the process of clearing is extremely simple. You just add (or subtract) the ordinary altitude corrections --no spherical trigonometry required (and you may remember a similar approach to clearing star-star distances published over a century later). What Nauticus didn't realize, quite ironically considering his complaint, is that the process of clearing lunars is so accepting of errors in the altitude of the Moon that a great many cases of clearing lunars, significantly distant from vertical circle conditions, can be converted to the ideal vertical circle case (which I outlined in a NavList post two years ago: http://fer3.com/x.aspx/05622). Even if the original observations are inconsistent, as above, within certain broad ranges the altitude of the Moon can be dropped and replaced by one calculated (by simple subtraction) under the assumption that the two bodies are actually aligned in a vertical circle. Then we do the clearing, and the results are every bit as good as if the full trigonometric solution had been worked out. As I joked in Mystic at June, I need a time machine to go back and fill everyone in on this... :)

-FER

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