The signs are obviously important, and you have to be careful about the direction of motion due to gravity. One thing though about my earlier attempt to explain the change in sign: if the angle is treated as negative and sin(-a) = -sin(a), why isn't the formula x = R(sin a - a)? Why does only the trig function's sign change, IOW? Well, wouldn't you have y = R(1 - cos a), so -y = -R(1 - cos a) = R(cos a - 1). So it looks like it isn't about negative angles after all (which kind of makes sense because the circle can move in either direction, generating both positive and negative angles). It's just an artifact of trig functions and I suppose circle symmetry. (??) Since gravity fixes the direction along s, Tach's claim that s has to be positive is clearly mistaken. Obviously s has an absolute length which is positive, but then ds/dt is a vector pointing in the same direction as gravity (-mg) at s = 0. Gosh amighty, how many mistakes can a person make? I think he's going for the record.
The above is as hilarious as your mistaken claims about the non-invariance proper time in the other thread. You are on a roll.
