NavList:

Herman, you noticed that the Hughes book says:
"At the same time rotate the instrument about the AXIS OF THE TELESCOPE "

Tsk-tsk! And good catch! Yes, that's the incorrect "modern" advice. This is certainly the earliest example of the flawed method that I have seen. It's conceivable that this is the original sin. :)

You also wrote:
"You learn in [ReedNavigation.com] workshops that this "General used advise" is usable to heights ~45° above that use the better method: Rotate round the index-mirror-star axis"

To clarify, I may have said that you won't go badly wrong using that "rotate about the scope axis" method for lower angles, but there's never any reason to apply it. Don't do it that way! Just do it right every time: rotate the instrument about the axis to the Sun or star, which is guaranteed by the simple visual test that the Sun or star should remain centered in the field of view as you rock back and forth. The horizon drops away below the celestial body except when the instrument is strictly vertical. And in that vertical position, if the angle is just right, the Sun will just kiss the horizon at the bottom of that rocking arc.

You asked:
"Why can I not find your better method, rotate round the index-mirror-star axis to keep the star in sight, in a book?"

Because the books all copy from each other? Because authors naively assume that there's no possibility of error in texts that have been copied from each other for so many years? And, among navigators, because they learn to accept and/or explain away the inaccuracy of their sights that arises from this simple "day one" flaw in their procedures? Celestial navigator are, by nature, independent-minded and sometimes pig-headed. I have had experienced navigators get testy with me when I show them the issue, and not just before they try it —when they're naturally skeptical— but after, probably because they're embarrassed that it works so well, and they missed out for many years.

Well, not all the books copy from each other. Or at least they don't do so naively. John Letcher in his "Self-Contained Celestial...", which we have discussed many times, gets it exactly right. I am sure there are other "modern" examples. You can also go back to early accounts of sextant use. There's one by Maskelyne himself over 200 years ago somewhere where he explains it reasonably well. 

Frank Reed
Clockwork Mapping / ReedNavigation.com
Conanicut Island USA