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I recently sent a post regarding finding compass variation on land using a
sextant, artificial horizon and compass. It was a method based on the
formula used by the explorer Augustus Gregory.

Hav(180dd-Z)= sech secL cosS cos(S-p)

Where
>Z = Azimuth angle (not Azimuth)
>h = altitude
>L = latitude
>p = polar distance (visible pole)
>S = 1/2 (h+L+p)

He was trying to solve for Z - the sun's Azimuth Angle. As both George
Huxtable and Doug Royer have pointed out this is an expression for the
azimuth angle of a body, to compare with its compass bearing and so obtain
the compass error assuming that because
the observer is on land there will be zero compass deviation.  In simple
terms, compass "error" becomes the difference between the calculated azimuth
angle of the sun and the compass bearing of the sun.

I showed these formulae to both Bruce Stark in the USA and George Bennet in
Australia many years ago before I became involved in this list and they
provided a lot of helpful analysis eg. George Bennett showed that the
following formula gives the same results as the one above (albeit you need a
calculator)

cos C = (cos c - cos a cos b)/ sin a sin b

Where
c = Polar Distance
a = co altitude
b= co latitude

We spent considerable time plugging numbers into these two formulae and
always got the same results. However this is for a restricted navigational
triangle based on the southern visible pole i.e. observer in the southern
hemisphere.

Bruce Stark pointed out at the time and I have since confirmed it that the
exact method including formula used by Gregory was in an 1804 Mackay's
navigator and a 1789 Moore as well as the formula I have already pointed to
in Bowditch.

Never satisfied with the restricted nature eo the solution provided by this
"visible pole" formula G bennet went looking for a "general universal
formula " derived from the cosine rule which would apply anywhere in the
world.. he came up with

 cos C = (sin dec - sin alt sin lat)/ cos alt cos lat

Where C = Azimuth of the body (0dd - 360dd)

George Bennett pointed out to me that because the cosine of an angle is
ambiguous the solution  is C for a morning observation or 360dd - C for an
afternoon observation. You must know beforehand.

Of course none of these formulae would have been capable of solution in the
field in the 19th century that's why they were broken down to logs and
haversines as follows:

Zn = hav (360DD- Zdd) = log sec L + log sec alt + log cos S + log cos (S-p)

Where Z = Azimuth Angle and Zn = Azimuth

Regards

Kieran Kelly
Sydney
Australia

   

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