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Alex E earlier quoted Maskelyne from 1772:

"Observers are commonly told, that in making the fore
observation they should move the index to bring the Sun
down to the part of the horizon directly beneath him, and turn
the quadrant about upon the axis of vision...
"I allow that this rule would be true, if a person could by sight
certainly know the part of the horizon beneath the Sun; but,
as this is impossible, the precept is incomplete.
Moreover, in taking the Sun's altitude in or near the zenith, this
rule entirely fails, and the best observers advise to
hold the quadrant vertical, and turn one's SELF ABOUT UPON THE HEEL,
stopping when the Sun glides along the horizon without cutting it:
and it is certain that this is a good rule in this case, and
capable with care of answering the intended purpose."

Many people (and the authors of many texts) still get this wrong. Maskelyne suggests that you swing the arc by turning it "about the axis of vision", which most people would consider to be the line through the telescope to the horizon. And I've seen many people do just that since rocking about the sight line to the horizon "appears" to give some sort of result. But actually the sextant has to be swung about the line of sight to the object. Notice that if you know the correct rule, then the second paragraph is nothing more than a special case of it since rotating the sextant about a vertical line means spinning about on your heels.

 

In historical logbooks, it is common to find references to Noon Sun sights where the Sun is nearly overhead. This is not really so difficult because they had a good idea of direction. When the Sun's altitude was suspected to be nearly 90 at Noon, they measured its altitude facing North and then facing South (may well have been done by two observers simultaneously).