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Clive Sutherland has suggested using the angle-in-the sky between pairs of
stars to correct the scale on Bill Murdoch's octant.

I've never tried measuring angles between stars, but wouldn't doubt that a
more experienced observer than I am could accomplish such a feat,
especially from a firm platform on land. All uncertainties connected with
observing the horizon would then, as Clive says, be removed.

I'm glad that Clive has corrected an initial error and agrees that
spherical geometry has to be used to work out the angles between stars.
However, another error needs to be corrected, when Clive states-

>Several angles on each pair would be measured and averaged,(just before to just
>after the calculated Meridian Pass) to ensure that the two stars of the
>pair will
>have the same Altitude.
>No correction will then be needed for Refraction.

It's true that the AMOUNT of the refraction correction would be the same
for two stars, both at the same altitude. However, that doesn't imply that
the angle between those stars is unaffected by the refraction. That would
be true only for two stars which were very nearly at the same azimuth. For
two stars at different azimuths, the trouble is that the refractions,
though the same in amount, are in quite different planes.

This becomes obvious when we take an extreme example. Consider two stars
which both, for a certain observer at a particular moment, have a
calculated altitude of 20 degrees (a zenith angle of 70 degrees). However,
refraction would cause both stars to appear to be a bit higher in the sky,
giving each (with a refraction taken from Norie at 2.6 minutes) a corrected
zenith angle of 69 degrees 57.4 minutes.

As yet, we haven't considered how far apart in azimuth these two stars
happen to be: now suppose it differs by 180 degrees, so the stars happen to
be in opposite directions in the sky (say, one to the East, one to the
West). The angle in the sky that the observer sees between these stars is
the arc between them that passes through his zenith, so it is 139 degrees
54.8 minutes (the two zenith angles added). So the refraction correction to
the angle between these two stars is 5.2 minutes, DOUBLE the refraction
correction for each star. If the difference in azimuth was less than 180
degrees, the refraction correction required would be less.

The conclusion has to be that for ANY two bodies, in comparing the measured
angle-in-the-sky with the tabulated value, the refraction for both bodies
has to be taken into account. And to do so the altitude of each body has to
be known, either by measurement or from tables. There's no advantage in
choosing two bodies with the same altitude. No great accuracy in the
altitude is required, because at any altitude, refraction is only a small
correction.

This is the problem that faced navigators of the late-18th and 19th century
when obtaining longitude from a Moon-Star or a Moon-Sun distance; that of
"clearing" the Lunar distance, or correcting it for the effects of
refraction of the two bodies and (for the Moon) the greater effect of
parallax. After that, the semidiameters had to be allowed-for.

The problem was that the Lunar Distance tables had to be suitable for use
from anywhere on the Earth, so they were compiled on the basis that the
angles between the centres of the bodies was shown, as if they had been
viewed from the centre of the Earth, and as if there were no atmosphere to
refract.

Hundreds of papers were written about methods of making these corrections,
and dozens of different calculation strategies were employed, using many
different sets of lookup tables. It tested to the limit the mathematical
ability of the navigators of the day, many of whom would go through the
routine motions without having the faintest idea what they were doing.
Really, the only calculation tool available to help was that of logarithms,
and as these can not deal with negative quanities, the trig. formulae had
to be twisted and deformed to fit in with that limitation. This only made
the process more abstruse to the ordinary navigator.

So a problem that we can solve in a moment with a pocket calculator became
an arcane mystery.

Getting back to Clive's proposal, I think he is overstating things a bit
when he says-

>Of course the final proof of the instrument calibration will be the accuracy of
>sights taken of the Sun etc as George Huxtable suggests. However this is
>the end
>point not the starting point of the exercise. All the corrections that we know
>and love,  necessary for a Sun sight e.g. Height of eye, Dip, Refraction
>Semi-diam, Irradiation,  all dilute the confidence in the calibration so
>obtained

Let's take these corrections one by one-

The total dip correction IS the height-of-eye correction; they are not
separate items. The height of eye should be easy to estimate to a foot or
so, and as the major component to dip is entirely geometrical (only the
Earth's diameter is involved) it can be calculated precvisely. There
remains a refraction-from-the-horizon contribution to the dip which can
vary markedly, but seldom exceeds two minutes, and this provides the
greates uncertaincy in sextant observation in otherwise-good conditions.
The purpose of a well-calibrated second sextant, alongside the octant under
test, would be to check whether abnormal refraction at the horizon is
occurring.

Provided observations are confined to angles over say 20 degrees altitude
then the refraction correction will be no greater than 2.6 minutes, and any
variation in the refraction from its standard value is unlikely to exceed
0.1 minutes. To calibrate smaller angles than 20 degrees, Clive's proposal
to measure angles-in the sky between two stars might well be appropriate.

Semi-diameter is a geometrical quantity that is known very precisely and
its accuracy is not in question.

Irradiation is a supposed effect within the human eye that makes bright
objects appear slightly larger than they actually are. It was not
universally accepted as real, though a suggested amount (for the Sun's
lower limb) was once included in tabulated values for its semidiameter. As
a result, an irradiation correction used to be required if measuring to the
Sun's upper limb. But in recent years, after the reality of the effect had
been questioned, all reference to irradiation has been dropped, as I
understand it.

So really, no strong reason remains to dismiss the use of Sun observations
in the calibration of Bill's octant.

I do hope that Bill appreciates all the attention that his proposal is
receiving. Whether or not he thinks the advice he is getting is appropriate
to his plans, he has raised some intriguing general questions.

George Huxtable.

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george@huxtable.u-net.com
George Huxtable, 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.
Tel. 01865 820222 or (int.) +44 1865 820222.
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