NavList:
A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding
From: Robin Stuart
Date: 2010 Sep 1, 11:17 -0700
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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