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Peter you wrote:
"any chance of some more  detail on this issue?"

Sure. I should say up-front that it's no big deal,  and you're probably
familiar with all of this but I'll just write a bit, and see  if it goes anywhere.

Angles ARE ratios. This is the fundamental  definition of an angle.

Almost all of us learn about angles at a very  early stage of our education
and we learn how to use angles, how to picture them  and compare them, and we
learn to count them in degrees. But the idea of  defining an angle is usually
left for much later, and by then students are so  familiar with angles that the
importance of the definition gets lost. An angle  is the ratio of the length
of an arc of a circle to its radius. That's all  --nothing more. And because
of this, angle have no units. Like any fraction, any  ratio, they are pure
numbers (no feet or meters or other physical units). So if  I have an object
that's a foot across and it's a hundred feet away from me, then  its angular size,
the angle it subtends, will be 1/100. Most of the time, people  refer to this
as an angle "measured in radians" but this is really an accident  of history
and education. An angle is just a ratio, and it's not really  "supposed to"
have any name after it.

Of course for those of us who  enjoy sextants and angle measuring, we like to
convert angles to the familiar  ancient sexagesimal terms. And to do that,
it's very convenient to memorize the  number of degrees or minutes or arcseconds
in a unit angle. If an angle is  0.001, for example, we can express it in
minutes of arc by multiplying by 3438  to get 3.4 minutes, or in degrees by
multiplying by 57.3, or in arcseconds by  multiplying by 206265.

If I want to know the angular size of my fist is  at arm's length, I just
divide 4 inches by 28 inches. That's an angle of 1/7  (or, multiplying by 57,
just about 8 degrees). And note that none of this has  anything to do with sines
and tangents and all that. We only need those when we  know that the geometry
is a true triangle.

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

   

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