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    Re: Impossible lunar example
    From: Douglas Denny
    Date: 2010 Sep 8, 12:24 -0700

    I have had some correspondence with George about this impossible lunars business. My last comments I post here which might stimulate further wider comment for the enlightenment of myself, if not others also confused.

    Dear George,

    Re. The other problem about impossible triangles. I have to admit total confusion as to what this is really all about with reference to the original old text.

    Yes you are of course quite right about the two specified altitudes (19 and 71) being impossible with a 103 degree length for the lunar distance; and yes we agree 180 degrees is the maximum length of any one side allowable in a spherical
    triangle.

    The impossibility also can be seen to follow from the fact that as the minimum
    distance between two points on a sphere is a great circle by definition, and the sides of all spherical triangles are great circles, then any two sides of a sph. triangle must together be greater than the third.

    So 71 +19 (Zenith distances - same as altitudes here) is 90 whereas the third is 103. Impossible by definition.

    (You can actually have the bit of string at 103 length between 19 and 71 if it is _curved_ on the surface - and then it is no longer a spherical triangle).

    What further confuses the issue for me, is that when I read the text when
    pointed to it, there is reference to Euclid 20 Book XII and there is no 20
    Book XII but there is 20 Book XI which gives the kind of statement refered to in the original text about three angles in a triangle, but refers to _solid_ angles in a triangle - which have nothing to do with it.
    This is where you seemed to me to be commenting on the text version
    regarding (bogus) solid angles. Excuse me please if this is not the case. I
    admit to confusion over the whole thing now.
    I conclude that yes the spherical triangle mentioned (19, 71, 103) is impossible; and suspect that the originator of the text confuses the problem with Euclid and solid angle propositions - which is strange for a teacher in spherical geometry. Or am I missing something altogether?


    Douglas Denny.
    Chichester. England.


    Original Posting:

    Douglas Denny wrote-
    "Which might be causing the confusion in the text with Janet Taylor because
    spherical angles are not solid angles."
    ------

    Yes they are. A spherical triangle on the Earth's surface subtends a solid
    angle at its centre. That is presumably the solid angle that the Euclid
    statement applies to.

    George.

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