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>>there's really only one solution to a spherical triangle, and it's been known for centuries. <<

This is interesting, Frank.  I have spent some time with Google on this, and can't figure out who discovered this solution, or when.  Do you have anything further on this?

>>After that it's just "trig identities" which are employed to make the work more convenient with <<

So people who came up with new formulations of these identities, which often resulted in different sets of published tables, were basically trying to address specific deficiencies they saw in (what were for them, in their time) existing tables?

Thanks for sharing your background on all this.

Bob