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Robin postulates a rather different spherical triangle to the "impossible"
one that we have been considering, joining the observer's zenith, and two
directions below the horizon, which are as far below it as the Moon and
star are above it.

I would suggest a somewhat more palatable way of looking at the same thing;
to leave the Moon and star in sight, where they actually are, above the
horizon, and construct a triangle between those directions and the
observer's nadir direction, directly beneath his feet, instead of his
zenith, above his head. Would Robin accept that as equivalent to what he
has been proposing?

Anyway, if Robin tries to construct such a triangle, on a globe, he will
fail. Why?

He wrote- "The triangle I describe has sides of 103, 109 and 161 degrees in
length, which does satisfy the condition that the sum of any two sides is
greater than the third, yet you seem to suggest that this does not
represent a real triangle and hence my argument is implausible. Can you
elaborate on what condition prevents this triangle from being constructed?"

Yes. It's true, as Robin states, that in this new triangle, the condition
that the sum of any two sides is greater than the third has not been
broken, but another condition, which must apply to all spherical triangles,
has, I'm not a geometer, and can't quote Euclid, but I am pretty certain
that a rule must exist to the effect that the sum of the sides of a
spherical triangle cannot exceed 360 degrees. When they add up to 360, the
three sides are strung out in line around a great circle, which divides the
sphere into two halves and which can't get any bigger. Robin's three sides
add up to 373 degrees. So his construction can't be done, no matter how
hard he tries.

George.

contact George Huxtable, at  george@hux.me.uk
or at +44 1865 820222 (from UK, 01865 820222)
or at 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.
----- Original Message -----
From: "Robin Stuart" 
To: 
Sent: Thursday, September 02, 2010 12:26 AM
Subject: [NavList] Re: Impossible lunar example


George,

      The question that I asked in my post may not be THE "question about
clearing such impossible triangles" but grant me at least that it is A
question!

I paraphrase it here:

Why is it that the application of spherical trigonometric identities to an
impossible triangle (one that could not be constructed, even in principle,
on the sphere) cannot be relied upon to show inconsistencies in the course
of calculation (a sin or cos greater than 1, or a square root of -1)?

The answer I provided was that the lunar distance clearing in this case
turns out to be equivalent to operations on a real triangle that can be
constructed on the sphere (even if, as noted, this triangle is not one that
would arise in clearing lunar distances in practice). A corollary is that
neither Maskelyne nor other practitioners can expect the mathematics to
alert them to the use of inconsistent starting values.

The triangle I describe has sides of 103, 109 and 161 degrees in length,
which does satisfy the condition that the sum of any two sides is greater
than the third, yet you seem to suggest that this does not represent a real
triangle and hence my argument is implausible. Can you elaborate on what
condition prevents this triangle from being constructed?

Regards,
Robin Stuart

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