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The Equation of Time is quite rigorously dealt with in W.M. Smart's book 
'Textbook on Spherical Astronomy'.

A formula is quoted based on the mean longitude of the Sun.
Being dependent on parameters of the Apparent Sun, because of Delta T it 
cannot be predicted in advance if great accuracy is required.
---------

For navigational use for meridional passage of the Sun a time to within one 
minute is quite satisfactory as the rate of change in altitude is small at 
maximum altitude for a good few minutes either side of Mer Passage.

To give some idea of this: herewith is a scan of a meridian passage of the Sun 
plotted as a graph from results found using my MkIXBM bubble sextant on Wed 
29th April this year.  I was playing around after calibrating the sextant 
(with a laser level down the garden and a target on a tree at about 20 Metres 
away), and used Mer Passage of the Sun as a method of checking accuracy (as 
it happened to be a nice sunny day).

There a few points worth noting:

1) the rate of change of altitude is very small near Mer Passage.
Change of altitude at Mer Passage within approx one minute of arc in altitude 
was approx from 11-46 to 12-07 in time i.e 21 minutes.  

2) Using a graph with best fit curve gives much more accurate results as the 
hyperbolic form of the curve can be deduced by eye, and using French curves, 
can be done more accurately than just averaging the results.

3) The variation of the personal observations of the observer and sextant 
both, can be deduced aproximately from the graph.  Using a graph with best 
fit curve the personal errors are largely eliminated leaving the sextant 
error mainly apparent.

You will note the variation around Mer Passage in the example (where there is 
very little change in altitude going on so the variations are largely 
personal errors plus sextant errors such as backlash in the gearing to the 
mirror) is about one minute of arc (i.e. plus or minus half a minute of arc) 
- which is exactly what I would have expected, as my sextant has backlash in 
the gear from altitude knob to mirror of around half a minute of arc in the 
reading.

4) a measure of accuracy can be fairly confidently made to around one minute 
of arc or better (depending on the backlash in the gearing) with care and 
lots of readings around Mer Passage.  I did this particular exercse for the 
fun of it on my own; but if a serious attempt is made for accuracy, another 
person present to take down the readings and measure time accurately as the 
observer concentrates on his business with the sextant would give much better 
results.

5) Within the limits of the rather slapdash observations, one can see the Mer 
Passage timing is exactly as predicted for the longitude of my position 
(obtained by GPS).

6)One can use the shadow of the Sun at exact Mer Passage time to establish an 
exact North/South line in your garden if you so wish.

If you do this accurately calculating the Mer Passage, using an accurate 
longitude, and using a theodolite,  you can establish N/S very accurately 
indeed.  It one of the methods used by surveyors.
(Stars/planets can be used as well of course).

7)By caclulating Latitude from the observed altitude at Mer Passage for the 
day/time by using declination (a very simple standard calculation - see 
graph) and comparing teh Latitude by observation with the known Latitude of 
my location (obtained by GPS)one can find directly the index error of the 
sextant to within the limits of the observation method (i.e. within about one 
minute of arc).
In this case the error was about 1.5 minute of arc. And so it proves in use.  
The line of position found by that sextant is usually very near plus or minus 
couple of minutes of arc under good conditions of observing, ...which is 
about what I normally expect as best possible for a bubble sextant.

Douglas Denny.
Chichester.  England.

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