are you sure the fan was supposed to come on just past half way on the temp gauge?
if you jump the fan wires to the battery and it come on, that is most likely fine, unless its noisy or binds up as mentioned..
then jump past the temp switch to test the whole system, this is actually easier, do it first. if that doesnt do it then check the relay and fan itself.. those are the three main parts to most cooling circuits.