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I wrote:
> With a sextant we
> are  measuring the altitude of point  B from point A so that means that  we
> know the angle in the triangle at point A,  let's call that  gamma.

Bill, you asked:
"Which triangle?  The small oblique or  large oblique?"

The only triangle I defined . You have point C  at the center of the
Earth, point A at height h above the Earth's surface, and  point B some distance
away at height H above the Earth's surface.Those three  points make a big
triangle ABC. Our GOAL is to find the angle in the triangle at  point C (since, if
we multiply that by the Earth's radius, we get the distance).  Now what do we
measure with our sextant or theodolite? Well, we're at point A,  and we measure
altitudes above the horizontal which is, by definition, the plane
perpendicular to side AC of the triangle. Now how is that related to the angle  at point
A in the big triangle (the angle "CAB")? Clearly that's just 90 degrees  plus
the measured altitude. So we KNOW the angle CAB. This is equivalent to the
measured parameter (or at minimum, it's related to the measured parameter by a
simple relationship). So we know the corner angle at point A (the observer),
which I named "gamma", and we are seeking the corner angle at point C (the
center of the Earth), which I named "phi". But, uh-oh, we're stuck with another
 angle --the one at point B. Wait... no we're not. Since it's a simple plane
triangle, all three angles must add up to 180 degrees. That means that the
angle  in the big triangle at point B is NECESSARILY equal to 180-(gamma+phi).
Ok so  far?? If you haven't drawn a picture of this yet, you can't possibly be
ok here   so please make sure you've got a picture of this. And if you
don't  want to draw your own picture, see image 488 in the archive (see  below).

Now we are in a position where we can apply the law of sines (the  ordinary
plane trig law of sines...) to the big triangle ABC. Set it up  as
sin("angle at A")/(R+H)=sin("angle at B")/(R+h).
And work  from there. You'll need to remember that
sin(a+b)=sin(a)*cos(b)+cos(a)*sin(b).

That's enough triangles for now, I  think!

I don't know if anyone will get a kick out of it or not, but if  you like
this kind of math/physics, I've photographed some of my notes. This is  RAW
material. You can download it at  www.HistoricalAtlas.com/lunars/refnotes.zip. It's
about 5.7 megabytes. By the  way, try to bookmark or record these addresses
with the "HistoricalAtlas" part  intact. Yes, it currently points to one of my
other addresses "clockwk.com" but  that's changing soon.

-FER
42.0N 87.7W, or 41.4N  72.1W.
www.HistoricalAtlas.com/lunars

   

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