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I fixed my calculation in Mathematica which had some small bugs. I
used the GHAand declination to one decimal place in minutes too.
Attached is a pdf printout of what I did.

It uses the spherical sine rule and Napier's analogies as I mentioned

On Tue, 12 May 2020 at 18:31, Geoffrey Kolbe
 wrote:
>
> What's with all this iteration...?
>
> I presented equations for finding LAT and LONG from the measured altitude Ho 
and measured azimuth Zo of a celestial body back in 2008 at the Mystic 
navigation conference when I gave a talk about navigating with a theodolite.. 
I am shocked nobody remembers!
>
> In this case, we have Ho = 54° 56', Zo = 120° 28' . The DEC for the sun was 
16° 46.9' and the GHA was 45° 51'
>
> Let MA = Sin-1(Sin Zo · Cos Ho / Cos DEC) = 31° 09'
>
> Since Zo is < 180, LHA = 360 - MA = 328° 51', from which we get a longitude 
of 77° West in the usual way
>
> We can find the latitude from:
>
> LAT = Sin-1[(Sin Ho·Sin DEC - Cos Ho·Cos DEC·Cos MA·Cos Zo)/(1 - (Cos DEC·Sin MA)2)] = 39° North
>
> The end
>
> Geoffrey Kolbe
>
> 

   

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