Hi Alex,
I tried what you suggested. I took the right triangle with corners at the north pole, at lattitude 0, longitude 0, and at lattitude 0 , longitude 90 degrees East. The length of each side is about 6000 miles, and when I plug them into your equation, it doesn't work.
I doubt that those numbers were defined much before Pythagoras. But if you go back another 50,000 years, I am pretty sure they weren't.
Anyway, the Pythagorean theorem only holds for triangles in a Euclidean plane, so you have to be sure that the one you are considering is really in such a plane. When it gets right down to it, I don't think we can really be sure we know of any exactly flat Euclidean planes, much less one with a precisely drawn triangle on it.
But all of that is just quibbling. What I refered to in my post was the formalism. I said that the mathematical formalism as it appears in the mathematical literature was invented by humans. I am quite sure that the Pythagorean Theorem could not be found in a math book prior to Pythagoras.
Now, the relationship that the theorem describes may have existed prior to Pythagoras. But there is still the question of whether or not there are any flat planes out there with nicely drawn triangles on them that might harbor that relationship among the lengths of their sides. That is, if the concept of length has any meaning outside of what human thinkers have given it.
Warm regards,
Paul
