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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. It is based on the formula

log (1-cos T) = log [cos(D-L) - sin H] + log sec L + log sec D

Sumner admits that method No. 1 has its own advantages. It's based on

log sin (T/2) = 1/2 * [ log sec H + log cosec PX + log cos s + log sin (s-H)].

Frankly, I don't know whether I would share Sumner's preference. Admittedly, no.
3 requires only 5 table entries, but in four different tables! XXIV, the natural
sines and cosines; XXVI, logarithms; XXVII, logarithms of secans;  and XXIII,
risings (being a versine table). No.1 (and No. 2 for that matter) can all be done
with table XXVII alone. De gustibus non est disputandum.

> 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.

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.

Herbert Prinz

P.S.

Those who want to read up on Sumner's method in Cotter, should replace the word
"observation" by "projection" in the quote on p.275. It makes much more sense
that way and reflects what Sumner actually wrote. Sumner also has "difficulty",
where Cotter says "defect". (4th ed. p.14).

   

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