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Herbert Prinz says-

>George Huxtable speculated:
>
>> There were several ways of deriving the longitude from a time-sight and the
>> latitude, but what he probably would use would be-
>>
>> cos (P/2) = sqrt( sin s sin (s - ZX) / (sin PZ sin PX))
>>
>> where P is the local hour angle, ZX is the Sun's zenith distance, PZ is the
>> co-lat, PX is the polar distance, and s = 1/2 (ZX + PZ + PX) .
>
>This would be Method No. 2 in Bowditch, of which I said in my previous
>post that
>Sumner didn't use it. As I said, he recommended (4th ed. p.18) Bowditch's third
>method, being the shortest one....

Yes, it was indeed speculation. I'm sorry for misunderstanding, and perhaps
contradicting, Herbert's original mailing, but with respect, that wasn't
what he said at all. The actual words in his first mailing, in response to
a question about how Sumner would have determined longitude, were-

"By chronometer and time sight, using one of the methods given in a contemporary
Bowditch (preferring No.3 over No.1, passing over No.2)."

I took that to mean that Herbert preferred (thought it most likely) that
Sumner would be using No.3 over No.1, and would pass over No.2, but clearly
that was my misunderstanding: It's clear now that he was referring to
Sumner's own preferences as stated in his 1843 pamphlet. As I said, I
didn't have access to Sumner's own account, having to rely on Cotter's
shortened version. I hope to read the full works within a few days.

But what took the usefulness out of Herbert's statement was his shorthand
reference to the methods in a "contemporary Bowditch", as Nos 1, 2, and 3,
without spelling out which was which. Unless a reader has access to an
appropriately old copy, he can have no idea which of Sumner's preferences
Herbert was indicating, in his statement above. (My modern Bowditch (vol 1,
1977) provides two formulae for the job, unnumbered).

Now Herbert has clarified that matter by identifying the three methods in
his latest mailing, and I thank him for that.

==========================================

For completeness, I will quote below some of Cotter's words about the
choice of formulae for deriving P, the angle at the pole between the
observer's meridian and that under the body (X) being observed.

Cotter starts by writing-

Cos P = (cos ZX - cos PZ.cos PX) cosec PZ. cosec PX

where PZ is the co-latitude of the observer,
ZX is the zenith distance of X,
PX is the polar distance of X

where PX is (90 + d) or (90 - d) (d being the declination of X) according
to whether latitude and declination have different or equivalent names
respectively.

He puts s = 1/2( PZ  + ZX + PX) (without specifically saying so), and goes
on to say-

"It is an easy matter to derive the following four formulas from the
fundamental cosine formula by expressing cos P in terms of sin P, sin
(P/2), cos (P/2) and tan (P/2) respectively.

sin P = 2 / (sin PX sin PX. SQR( sin s sin (s-ZX) sin (s-PZ) sin(s-PX))

sin (P/2) = SQR (sin (s-PZ) sin (s-PX) / (sin PZ sin PX))

cos (P/2) = SQR (sin s sin (s-ZX) / (sin PZ sin PX)) [note that in this
expression I have deleted an intrusive subscript "2", which is presumably a
typographical error- GH]

tan (P/2) = SQR (sin (s-PZ) sin (s-PX) / (sin s sin (s-ZX)))"

[note that these equations have here been re-jigged a bit to fit in with
email typography, and SQR means "square root of" - GH]

Cotter goes on to say-

"All of these formulae are adapted for logarithmic computation. To which of
the four preference ought to be given over the others should depend on the
value of the angle P. It can be demonstrated that the first is suitable for
cases in which P is near 90 degrees. It is expedient to use the second or
fourth when P is acute; and to use the third when P considerably exceeds 90
degrees." End of quote from Cotter.

We have learned to take much of what Cotter says with a pinch of salt, and
his statement that "the first is suitable for cases in which P is near 90
degrees" seems highly dubious, because deriving P from sin P, when P is
near 90 degrees, is both inaccurate and ambiguous. I suggest that statement
should have read- "the first is UNsuitable for cases in which P is near 90
degrees".

The formula involving cos (P/2) had achieved a long ancestry even by
Sumner's day. It formed part of Robert Bishop's blank form for deriving
longitude from a lunar, printed from an engraving in 1768, so by Sumner's
time it was all of 70 years old.

===========================

Thanks to Herbert Prinz for locating further errors in Cotter.

Herbert asked what a take to be a rhetorical question- "How common was it
to do a time sight at all?". What was he implying by that? Surely any
mariner who wanted to determine his longitude would have to take some sort
of time-sight, involving two altitudes. It was required for any mariner who
wished to proceed beyond latitude sailing.

Finally, in response to my observation-

>> It's an interesting question, why it had to wait until 1837 before mariners
>> had the commonsense to realise that an oblique position line could be drawn
>> from a single altitude of any body with a known position in the sky.
>> Looking back, it seems such an obvious step.

Herbert responded-

>For one, it had to wait for the chronometer. It would seem to me that the
>method
>was proposed reasonably soon after the general availability of this instrument.

I don't see that it had to wait for the chronometer (which, it's true, was
only then beginning to percolate down within the reach of ordinary
mariners, outside the elite ranks of the Royal Navy and the East India
company). The cleverer mariners had been finding their time using lunars
for the previous 70 years. There's no difference between a time measured by
a lunar and a time measured by a chronometer (except for its precision,
perhaps). So wouldn't position-line navigation have been just as useful to
a navigator that had obtained his time from a lunar, as to a chronometer
user?

George Huxtable.


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