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George Huxtable wrote:
>
> But I remain puzzled about Toynbee's discussion of the implications of the
> differences between those observations. If Paul's explanation is correct,
> they relate to nothing more than different assessments of the apparent
> diameter of the Sun, but Toynbee seems to think there's more substance in it
> than that.

Actually, I think the differences between the members of each pair are
due to personal error in making the coincidence. Toynbee says, "On
referring to the above table it will be noticed that in Mr. Quihampton's
case, the [Sun W Moon] lunars invariably give the greater, but in mine
almost as invariably the lesser error. This may arise from a constant
error in the observer, or his instrument; but the means of our [Sun E
Moon] and [Sun W Moon] errors nearly agree."

He also says, "I think it has also proved that there is a constant
character in lunars taken by the same observer, using the same
instrument in the same state; whereby any person having practised this
method for a short time, may readily show how much a single set of
lunars gives the Greenwich time in error. For instance, Mr. Quihampton's
[Sun W Moon] and [Sun E Moon] lunars differ in the Greenwich time about
one minute and twenty seconds; now the half of this, or forty seconds of
time, may be called his constant error; mine differ about twenty-nine
seconds, so that my constant error is fifteen seconds of time, which
applied to any single set of observations will give the Greenwich time
very nearly correct: hence an observer having ascertained his constant
error, has the true longitude from a single set, taken either [Sun E
Moon] or [Sun W Moon]."


Incidentally, Charles Shadwell's book "Notes on the Management of
Chronometers and the Measurement of Meridian Distances" (1855) is on
Google books. (A "meridian distance" is the longitude difference between
two places.)
http://books.google.com/books?id=hjMDAAAAQAAJ

Cotter called this work a "classic". At the Google site I noticed it was
republished in 2008. The MNRAS reviewed the original back in 1855:

http://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?journal=MNRAS&year=%3f%3f%3f%3f&volume=..15&letter=.&db_key=PRE&page_ind=195&plate_select=NO&data_type=GIF&type=SCREEN_GIF&classic=YES&high=49b0469be600586

Shadwell throws some light on the question of how many chronometers were
carried: "In Her Majesty's Navy it is usually customary to furnish every
ship with one chronometer. If the captain supplies a private one in
addition, then the Government give another, so as to make three
chronometers in all. It is argued, that if a ship have but one
chronometer, it would be unwise implicitly to trust it; and, therefore,
great caution is necessary in navigation. If the ship had two, and they
happened to differ, it would be impossible to tell which was right. If,
however, she had three, the coincidence of any two of them would throw a
strong probability on the truth of their results; while the mean of the
three may probably be more safely relied on than any one of them taken
singly: added to which, the examination of their intercomparisons, as
pointed out above, gives the means of detecting which of the three is
irregular."

An appendix of his book has the chronometer rate log from HMS Fly during
four years exploring the Australian coast. That ship carried 12
chronometers.

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