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Good day to you Jim,

(1) - As regards your query about the way Captain Cook & Co derived the apparent time of the eclipse, they certainly had to iterate when back to England - as I did, see (2) hereunder. They could rely on a carefully recorded upper Sun height for the beginning of the eclipse from which they could derive the Sun geocentric height. Knowing both Geocentric Height, Latitude and with a good starting then iterating value for the Sun declination, they were able to immediately derive the Sun Local Hour Angle - which is identical to their "apparent time". They did so for the beginning of the eclipse and then for the end of the eclipse.

How did they exactly record the time "one minute after the beginning of the eclipse" is something I cannot figure out. But as regards deriving a "local time" from the Geocentric Height, Declination and Latitude, we exactly know how they proceeded, or at least we know which formula they using : the well known one.

Now, unfortunately the Royal Society original report by Mr J. Bevis is packed up with typos, (you even added one on your own when first mentioning the Latitude) . Hence unfortunately we are dealing here with significantly corrected / reconstructed data.

Nonetheless, we can still study this eclipse thanks to your sagacity Frank.

(2) - I have been working on this full example starting with only the date (Aug 05th, 1766, TT-UT = +15.6s)  , the total eclipse duration (i.e. a change of 43.108333° in the local time) , and the Latitude (N47°36'19"). All these data were no doubt fully known and recorded.

Then by approximations, and from own software - accurate to +/- 0.2" for the Sun and +/- 5" for the Moon - for fixed Latitude at N47°36'19":

At 50°W , eclipse spanned over a LHA change of 40.85433°

At 55°W , eclipse spanned over a LHA change of 42.39653°

At 56°W , eclipse spanned over a LHA change of 42.68673°

At 57°W , eclipse spanned over a LHA change of 42.97134°

At 57°28'W , eclipse spanned over a LHA change of 43.10175°, and finally :

At 57°29'26" W, the eclipse spanned over a LHA change of 43.10822° which for the sake of the drill we will consider as adequately matching the 43.10833° benchmark.

Therefore, from the given data, I am offering the best following solution :

Beginning of Eclipse at 16h42m43s UT, end of Eclipse at 19h35m09s at position : N47°36'19" / W057°29'26" ,

This is only 6 arc minutes off the current Google Earth position very close to W057°35' : a performance for these times.

How about the Sun upper limb zenith distance ? I have not used it so far.

Apparent UL zenith distance 31°57' is the same as apparent height at 58°03' with height of eye equal to 0'.

If I compute the Apparent UL height for the "begining of the eclipse + 1minute of time", i.e. for 16h43m43.3s , I find 58°00'11" with a diffference of almost 3' which is not convincing. They were able to perform much better measurements especially from land.

Latitude was already known then to just a few arc seconds ...

On the other hand if I compute the UL height for the beginning time at 16h42m43,3 s , then I find 58°03'27", almost identical to the published 58°03'.

So ... the "one minute after beginning of the eclipse" story might be challenged here, especially since this overall article was certainly not proof-read by a knowledgeable reviewer.

We are unfortunately working from unreliable historical data which certainly were not proof-read by an adequately competent reviewer.

Nonetheless, and thanks to the "most reasonable/probable/realistic" corrections brought by Frank, and through modern software we are able to position ourselves with an end longitude error of about 6 arc minutes which is a remarkable achievement.

Kermit