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The question about clearing such impossible triangles is not in the 
validity of the mathematical manipulations but whether the end-result has 
any physical meaning.

Robin invites us to accept the meaningfulness of a "possible" triangle in 
which-

LD = 103
ZDmoon = 180 - 71 = 109 (or 19º below the horixon)
ZDsun = 180 - 19 = 161 (or 71º below the horizon).

Even if we inhabited a see-through World, in which such directions could be 
observed, I fail to see any way in which the angle between them could reach 
a lunar distance of 103º. Robin's alternative is even less plausible than 
the original triangle.

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: Wednesday, September 01, 2010 7:17 PM
Subject: [NavList] Re: Impossible lunar example


One way to explain why clearing the lunar distance on the impossible 
triangle does not fail and proceeds as, George put it, "without meeting 
along the way a sin or cos greater than 1, or a square root of -1" is that 
it is operationally equivalent to clearing the lunar distance on a 
particular "possible" triangle.

To see this, consider clearing the lunar distance using the cosine formula 
of spherical trigonometry in the form
cos(LD) = cos(ZDmoon)*cos(ZDsun) + sin(ZDmoon)*sin(ZDsun)*cos(Z)
LD = lunar distance
ZDmoon = Moon's zenithal distance
ZDsun = Sun's zenithal distance
Z = Sun and Moon azimuth difference

For the example of the impossible triangle
LD = 103
ZDmoon = 71
ZDsun = 19

Note however that

cos(ZDmoon)*cos(ZDsun) = cos(180 - ZDmoon)*cos(180 - ZDsun)
sin(ZDmoon)*sin(ZDsun) = sin(180 - ZDmoon)*sin(180 - ZDsun)

And hence clearing the lunar distance on the impossible triangle is 
operationally equivalent to clearing the lunar distance on a triangle with 
sides
LD = 103
ZDmoon = 180 - 71 = 109
ZDsun = 180 - 19 = 161
which is an entirely "possible" triangle. Of course with ZD's > 90 this is 
not one that would arise in practice but the operations performed in the 
course of lunar distance clearing are all trigonometrically valid and 
geometrically meaningful even though a positive correction (refraction + 
dip etc.) applied to the ZD's of the impossible triangle corresponds to a 
negative correction applied to the sides of its "possible" cousin,

Robin Stuart


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