NavList:

I admit to not following this thread so might be talking out of turn: but do not understand why you call this triangle impossible? It is not an impossible spherical triangle at all.

The limiting case for a spherical triangle is where one side is 180 degrees in which case it occupies a hemisphere.

So:
each side or angle of a spherical triangle is < 180
the sum of the three sides is between 0 and 360
the sum of the three angles is between 180 and 540
the area of of any sph. triangle must be less than 2.pi.R^2

By co-incidence I have just recently programmed my HP50 for "clearing" Lunar distances with the standard formula which I used as a basis from Cotter's 'History of Nautical Astronomy' p.209

cos D = sin S sin M + (cos d - sin s sin m )(cos S cos M)/(cos s cos m)

D=lunar distance; S = true altitude star; M = true altitude of Moon; s = apparent alt star; m = apparent altitude Moon; d = apparent lunar distance.

Putting your figures into my calculator for clearing a lunar distance (it includes refraction correction) gives:

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

Cleared distance = 103.061 degrees.

Douglas Denny.
Chichester. England.

======================

Original Message:-

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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