Man, I COMPLETELY understand, unfortunately I have no idea. You see, mine is doing the exact same thing (65 200 cid 3speed). EXACTLY the same situation. I replaced the solenoid (twice), the grounds, the battery, the ignition switch, and the starter - NO difference. I turn the key and the starter just whirrs (technical term there for all you laymen) but never touches the engine. I tried a different starter, same thing.
The problem with our problem is that it doesn't even exist! The whole starting mechanism of that car consists of the following: Good battery charge, good battery ground, solenoid, and the starter's bendix drive drives out the gear which then hits the fly wheel and starts the engine... that's it - mine has/does ALL of that perfects and still doesn't start (which is, as I said, not even technically possible, yet here we are)
Now that I am done ranting and raving, in case your problem is fixable, here are the things you need to look at:
* That your battery has enough charge (it takes a lot) - trying jumping off the solenoid to check (make sure the car is NOT in gear when you do this!)
* That your starter is good - in the right amp range and that the bendix gear is running out all the way (you can jump your starter off - OUTside of your car - with a pair of jumper cables to check that)
* Check the ground wire on your battery
* look at your flywheel to make sure it has no broken teeth (major problem here - that is most likely the problem for most people)
You can turn your flywheel to try to get a good spot that will catch (put your car in gear - turned off - and try to push it backwards - rocking it backwards- this will turn your flywheel).
Also, where did you get your starter and what is the part number on it (a 3131?) I would really like to know that.
PLEASE PLEASE PLEASE let me know if you find ANYthing out!